{"id":73,"date":"2021-03-25T02:20:52","date_gmt":"2021-03-25T02:20:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-2\/"},"modified":"2022-01-03T18:27:55","modified_gmt":"2022-01-03T18:27:55","slug":"basics-of-differential-equations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-2\/","title":{"raw":"Problem Set: Basics of Differential Equations","rendered":"Problem Set: Basics of Differential Equations"},"content":{"raw":"

Determine the order of the following differential equations.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]{y}^{\\prime }+y=3{y}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"124732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124732\"][latex]1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]{\\left({y}^{\\prime }\\right)}^{2}={y}^{\\prime }+2y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>[latex]y^{\\prime\\prime\\prime}+y^{\\prime\\prime}{y}^{\\prime }=3{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"756272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"756272\"][latex]3[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]{y}^{\\prime }=y^{\\prime\\prime}+3{t}^{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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5.\u00a0<\/strong>[latex]\\frac{dy}{dt}=t[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"570545\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"570545\"][latex]1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]\\frac{dy}{dx}+\\frac{{d}^{2}y}{d{x}^{2}}=3{x}^{4}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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7.\u00a0<\/strong>[latex]{\\left(\\frac{dy}{dt}\\right)}^{2}+8\\frac{dy}{dt}+3y=4t[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"592840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"592840\"][latex]1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Verify that the following functions are solutions to the given differential equation.<\/p>\r\n\r\n

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8.\u00a0<\/strong>[latex]y=\\frac{{x}^{3}}{3}[\/latex] solves [latex]{y}^{\\prime }={x}^{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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9.\u00a0<\/strong>[latex]y=2{e}^{\\text{-}x}+x - 1[\/latex] solves [latex]{y}^{\\prime }=x-y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]y={e}^{3x}-\\frac{{e}^{x}}{2}[\/latex] solves [latex]{y}^{\\prime }=3y+{e}^{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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11.\u00a0<\/strong>[latex]y=\\frac{1}{1-x}[\/latex] solves [latex]{y}^{\\prime }={y}^{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]y={e}^{\\frac{{x}^{2}}{2}}[\/latex] solves [latex]{y}^{\\prime }=xy[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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13.\u00a0<\/strong>[latex]y=4+\\text{ln}x[\/latex] solves [latex]x{y}^{\\prime }=1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]y=3-x+x\\text{ln}x[\/latex] solves [latex]{y}^{\\prime }=\\text{ln}x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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15.\u00a0<\/strong>[latex]y=2{e}^{x}-x - 1[\/latex] solves [latex]{y}^{\\prime }=y+x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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16.\u00a0<\/strong>[latex]y={e}^{x}+\\frac{\\sin{x}}{2}-\\frac{\\cos{x}}{2}[\/latex] solves [latex]{y}^{\\prime }=\\cos{x}+y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]y=\\pi {e}^{\\text{-}\\cos{x}}[\/latex] solves [latex]{y}^{\\prime }=y\\sin{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Verify the following general solutions and find the particular solution.<\/p>\r\n\r\n

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18.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=4{x}^{2}[\/latex] that passes through [latex]\\left(-3,-30\\right)[\/latex], given that [latex]y=C+\\frac{4{x}^{3}}{3}[\/latex] is a general solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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19.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{3}[\/latex] that passes through [latex]\\left(1,4.75\\right)[\/latex], given that [latex]y=C+\\frac{3{x}^{4}}{4}[\/latex] is a general solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"628422\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"628422\"][latex]y=4+\\frac{3{x}^{4}}{4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{2}y[\/latex] that passes through [latex]\\left(0,12\\right)[\/latex], given that [latex]y=C{e}^{{x}^{3}}[\/latex] is a general solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=2xy[\/latex] that passes through [latex]\\left(0,\\frac{1}{2}\\right)[\/latex], given that [latex]y=C{e}^{{x}^{2}}[\/latex] is a general solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"561411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"561411\"][latex]y=\\frac{1}{2}{e}^{{x}^{2}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }={\\left(2xy\\right)}^{2}[\/latex] that passes through [latex]\\left(1,-\\frac{1}{2}\\right)[\/latex], given that [latex]y=-\\frac{3}{C+4{x}^{3}}[\/latex] is a general solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }{x}^{2}=y[\/latex] that passes through [latex]\\left(1,\\frac{2}{e}\\right)[\/latex], given that [latex]y=C{e}^{\\frac{\\text{-}1}{x}}[\/latex] is a general solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"242113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"242113\"][latex]y=2{e}^{\\frac{\\text{-}1}{x}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>Find the particular solution to the differential equation [latex]8\\frac{dx}{dt}=-2\\cos\\left(2t\\right)-\\cos\\left(4t\\right)[\/latex] that passes through [latex]\\left(\\pi ,\\pi \\right)[\/latex], given that [latex]x=C-\\frac{1}{8}\\sin\\left(2t\\right)-\\frac{1}{32}\\sin\\left(4t\\right)[\/latex] is a general solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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25.\u00a0<\/strong>Find the particular solution to the differential equation [latex]\\frac{du}{dt}=\\tan{u}[\/latex] that passes through [latex]\\left(1,\\frac{\\pi }{2}\\right)[\/latex], given that [latex]u={\\sin}^{-1}\\left({e}^{C+t}\\right)[\/latex] is a general solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"550576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"550576\"][latex]u={\\sin}^{-1}\\left({e}^{-1+t}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>Find the particular solution to the differential equation [latex]\\frac{dy}{dt}={e}^{\\left(t+y\\right)}[\/latex] that passes through [latex]\\left(1,0\\right)[\/latex], given that [latex]y=\\text{-}\\text{ln}\\left(C-{e}^{t}\\right)[\/latex] is a general solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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27.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }\\left(1-{x}^{2}\\right)=1+y[\/latex] that passes through [latex]\\left(0,-2\\right)[\/latex], given that [latex]y=C\\frac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex] is a general solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"789877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"789877\"][latex]y=-\\frac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following problems, find the general solution to the differential equation.<\/p>\r\n\r\n

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28.\u00a0<\/strong>[latex]{y}^{\\prime }=3x+{e}^{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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29.\u00a0<\/strong>[latex]{y}^{\\prime }=\\text{ln}x+\\tan{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"462506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"462506\"][latex]y=C-x+x\\text{ln}x-\\text{ln}\\left(\\cos{x}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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30.\u00a0<\/strong>[latex]{y}^{\\prime }=\\sin{x}{e}^{\\cos{x}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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31.\u00a0<\/strong>[latex]{y}^{\\prime }={4}^{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"997965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997965\"][latex]y=C+\\frac{{4}^{x}}{\\text{ln}\\left(4\\right)}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]{y}^{\\prime }={\\sin}^{-1}\\left(2x\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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33.\u00a0<\/strong>[latex]{y}^{\\prime }=2t\\sqrt{{t}^{2}+16}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"544555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"544555\"][latex]y=\\frac{2}{3}\\sqrt{{t}^{2}+16}\\left({t}^{2}+16\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]{x}^{\\prime }=\\text{coth}t+\\text{ln}t+3{t}^{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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35.\u00a0<\/strong>[latex]{x}^{\\prime }=t\\sqrt{4+t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"425862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425862\"][latex]x=\\frac{2}{15}\\sqrt{4+t}\\left(3{t}^{2}+4t - 32\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]{y}^{\\prime }=y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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37.\u00a0<\/strong>[latex]{y}^{\\prime }=\\frac{y}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"105971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"105971\"][latex]y=Cx[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the following initial-value problems starting from [latex]y\\left(t=0\\right)=1[\/latex] and [latex]y\\left(t=0\\right)=-1[\/latex]. Draw both solutions on the same graph.<\/p>\r\n\r\n

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38.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2t[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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39.\u00a0<\/strong>[latex]\\frac{dy}{dt}=\\text{-}t[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"353475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"353475\"][latex]y=1-\\frac{{t}^{2}}{2},y=-\\frac{{t}^{2}}{2}-1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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41.\u00a0<\/strong>[latex]\\frac{dy}{dt}=\\text{-}y[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"498655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"498655\"][latex]y={e}^{\\text{-}t},y=\\text{-}{e}^{\\text{-}t}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the following initial-value problems starting from [latex]{y}_{0}=10[\/latex]. At what time does [latex]y[\/latex] increase to [latex]100[\/latex] or drop to [latex]1?[\/latex]<\/p>\r\n\r\n

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43.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4t[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"257066\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257066\"][latex]y=2\\left({t}^{2}+5\\right),t=3\\sqrt{5}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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45.\u00a0<\/strong>[latex]\\frac{dy}{dt}=-2y[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"359360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359360\"][latex]y=10{e}^{-2t},t=-\\frac{1}{2}\\text{ln}\\left(\\frac{1}{10}\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>[latex]\\frac{dy}{dt}={e}^{4t}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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47.\u00a0<\/strong>[latex]\\frac{dy}{dt}={e}^{-4t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"544392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"544392\"][latex]y=\\frac{1}{4}\\left(41-{e}^{-4t}\\right)[\/latex], never[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from [latex]y\\left(t=0\\right)=-10[\/latex] to [latex]y\\left(t=0\\right)=10[\/latex] increasing by [latex]2[\/latex]. Is there some critical point where the behavior of the solution begins to change?<\/p>\r\n\r\n

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48. [T]<\/strong> [latex]{y}^{\\prime }=y\\left(x\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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49. [T]<\/strong> [latex]x{y}^{\\prime }=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"925345\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"925345\"]Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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50. [T]<\/strong> [latex]{y}^{\\prime }={t}^{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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51. [T]<\/strong> [latex]{y}^{\\prime }=x+y[\/latex] (Hint:<\/em> [latex]y=C{e}^{x}-x - 1[\/latex] is the general solution)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"605384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"605384\"]Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

52. [T]<\/strong> [latex]{y}^{\\prime }=x\\text{ln}x+\\sin{x}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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53.\u00a0<\/strong>Find the general solution to describe the velocity of a ball of mass [latex]1\\text{lb}[\/latex] that is thrown upward at a rate [latex]a[\/latex] ft\/sec.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"150056\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"150056\"][latex]v\\left(t\\right)=-32t+a[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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54.\u00a0<\/strong>In the preceding problem, if the initial velocity of the ball thrown into the air is [latex]a=25[\/latex] ft\/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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55.\u00a0<\/strong>You throw two objects with differing masses [latex]{m}_{1}[\/latex] and [latex]{m}_{2}[\/latex] upward into the air with the same initial velocity [latex]a[\/latex] ft\/s. What is the difference in their velocity after [latex]1[\/latex] second?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"450658\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"450658\"][latex]0[\/latex] ft\/s[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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56. [T]<\/strong> You throw a ball of mass [latex]1[\/latex] kilogram upward with a velocity of [latex]a=25[\/latex] m\/s on Mars, where the force of gravity is [latex]g=-3.711[\/latex] m\/s2<\/sup>. Use your calculator to approximate how much longer the ball is in the air on Mars than on Earth, where [latex]g= -9.8 m\/s^{2}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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57. [T]<\/strong> For the previous problem, use your calculator to approximate how much higher the ball went on Mars, where [latex]g= -9.8 m\/s^{2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"790883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790883\"][latex]4.86[\/latex] meters[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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58. [T]<\/strong> A car on the freeway accelerates according to [latex]a=15\\cos\\left(\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of [latex]51[\/latex] mph. After [latex]40[\/latex] minutes of driving, what is the driver\u2019s velocity?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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59. [T]<\/strong> For the car in the preceding problem, find the expression for the distance the car has traveled in time [latex]t[\/latex], assuming an initial distance of [latex]0[\/latex]. How long does it take the car to travel [latex]100[\/latex] miles? Round your answer to hours and minutes.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"334555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"334555\"][latex]x=50t-\\frac{15}{{\\pi }^{2}}\\cos\\left(\\pi t\\right)+\\frac{3}{{\\pi }^{2}},2[\/latex] hours [latex]1[\/latex] minute[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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60. [T]<\/strong> For the previous problem, find the total distance traveled in the first hour.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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61.\u00a0<\/strong>Substitute [latex]y=B{e}^{3t}[\/latex] into [latex]{y}^{\\prime }-y=8{e}^{3t}[\/latex] to find a particular solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"730642\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730642\"][latex]y=4{e}^{3t}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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62.\u00a0<\/strong>Substitute [latex]y=a\\cos\\left(2t\\right)+b\\sin\\left(2t\\right)[\/latex] into [latex]{y}^{\\prime }+y=4\\sin\\left(2t\\right)[\/latex] to find a particular solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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63.\u00a0<\/strong>Substitute [latex]y=a+bt+c{t}^{2}[\/latex] into [latex]{y}^{\\prime }+y=1+{t}^{2}[\/latex] to find a particular solution.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"853899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"853899\"][latex]y=3 - 2t+{t}^{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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64.\u00a0<\/strong>Substitute [latex]y=a{e}^{t}\\cos{t}+b{e}^{t}\\sin{t}[\/latex] into [latex]{y}^{\\prime }=2{e}^{t}\\cos{t}[\/latex] to find a particular solution.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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65.\u00a0<\/strong>Solve [latex]{y}^{\\prime }={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex] and solve [latex]{y}^{\\prime }=1[\/latex] with the same initial condition. As [latex]k[\/latex] approaches [latex]0[\/latex], what do you notice?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"688317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688317\"][latex]y=\\frac{1}{k}\\left({e}^{kt}-1\\right)[\/latex] and [latex]y=x[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Determine the order of the following differential equations.<\/p>\n

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1.\u00a0<\/strong>[latex]{y}^{\\prime }+y=3{y}^{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]{\\left({y}^{\\prime }\\right)}^{2}={y}^{\\prime }+2y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>[latex]y^{\\prime\\prime\\prime}+y^{\\prime\\prime}{y}^{\\prime }=3{x}^{2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>[latex]{y}^{\\prime }=y^{\\prime\\prime}+3{t}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]\\frac{dy}{dt}=t[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>[latex]\\frac{dy}{dx}+\\frac{{d}^{2}y}{d{x}^{2}}=3{x}^{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]{\\left(\\frac{dy}{dt}\\right)}^{2}+8\\frac{dy}{dt}+3y=4t[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Verify that the following functions are solutions to the given differential equation.<\/p>\n

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8.\u00a0<\/strong>[latex]y=\\frac{{x}^{3}}{3}[\/latex] solves [latex]{y}^{\\prime }={x}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>[latex]y=2{e}^{\\text{-}x}+x - 1[\/latex] solves [latex]{y}^{\\prime }=x-y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]y={e}^{3x}-\\frac{{e}^{x}}{2}[\/latex] solves [latex]{y}^{\\prime }=3y+{e}^{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>[latex]y=\\frac{1}{1-x}[\/latex] solves [latex]{y}^{\\prime }={y}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]y={e}^{\\frac{{x}^{2}}{2}}[\/latex] solves [latex]{y}^{\\prime }=xy[\/latex]<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]y=4+\\text{ln}x[\/latex] solves [latex]x{y}^{\\prime }=1[\/latex]<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]y=3-x+x\\text{ln}x[\/latex] solves [latex]{y}^{\\prime }=\\text{ln}x[\/latex]<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]y=2{e}^{x}-x - 1[\/latex] solves [latex]{y}^{\\prime }=y+x[\/latex]<\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>[latex]y={e}^{x}+\\frac{\\sin{x}}{2}-\\frac{\\cos{x}}{2}[\/latex] solves [latex]{y}^{\\prime }=\\cos{x}+y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]y=\\pi {e}^{\\text{-}\\cos{x}}[\/latex] solves [latex]{y}^{\\prime }=y\\sin{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n

Verify the following general solutions and find the particular solution.<\/p>\n

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18.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=4{x}^{2}[\/latex] that passes through [latex]\\left(-3,-30\\right)[\/latex], given that [latex]y=C+\\frac{4{x}^{3}}{3}[\/latex] is a general solution.<\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{3}[\/latex] that passes through [latex]\\left(1,4.75\\right)[\/latex], given that [latex]y=C+\\frac{3{x}^{4}}{4}[\/latex] is a general solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=4+\\frac{3{x}^{4}}{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=3{x}^{2}y[\/latex] that passes through [latex]\\left(0,12\\right)[\/latex], given that [latex]y=C{e}^{{x}^{3}}[\/latex] is a general solution.<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }=2xy[\/latex] that passes through [latex]\\left(0,\\frac{1}{2}\\right)[\/latex], given that [latex]y=C{e}^{{x}^{2}}[\/latex] is a general solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{1}{2}{e}^{{x}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }={\\left(2xy\\right)}^{2}[\/latex] that passes through [latex]\\left(1,-\\frac{1}{2}\\right)[\/latex], given that [latex]y=-\\frac{3}{C+4{x}^{3}}[\/latex] is a general solution.<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }{x}^{2}=y[\/latex] that passes through [latex]\\left(1,\\frac{2}{e}\\right)[\/latex], given that [latex]y=C{e}^{\\frac{\\text{-}1}{x}}[\/latex] is a general solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=2{e}^{\\frac{\\text{-}1}{x}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>Find the particular solution to the differential equation [latex]8\\frac{dx}{dt}=-2\\cos\\left(2t\\right)-\\cos\\left(4t\\right)[\/latex] that passes through [latex]\\left(\\pi ,\\pi \\right)[\/latex], given that [latex]x=C-\\frac{1}{8}\\sin\\left(2t\\right)-\\frac{1}{32}\\sin\\left(4t\\right)[\/latex] is a general solution.<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>Find the particular solution to the differential equation [latex]\\frac{du}{dt}=\\tan{u}[\/latex] that passes through [latex]\\left(1,\\frac{\\pi }{2}\\right)[\/latex], given that [latex]u={\\sin}^{-1}\\left({e}^{C+t}\\right)[\/latex] is a general solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]u={\\sin}^{-1}\\left({e}^{-1+t}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>Find the particular solution to the differential equation [latex]\\frac{dy}{dt}={e}^{\\left(t+y\\right)}[\/latex] that passes through [latex]\\left(1,0\\right)[\/latex], given that [latex]y=\\text{-}\\text{ln}\\left(C-{e}^{t}\\right)[\/latex] is a general solution.<\/div>\n<\/div>\n<\/div>\n
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27.\u00a0<\/strong>Find the particular solution to the differential equation [latex]{y}^{\\prime }\\left(1-{x}^{2}\\right)=1+y[\/latex] that passes through [latex]\\left(0,-2\\right)[\/latex], given that [latex]y=C\\frac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex] is a general solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=-\\frac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following problems, find the general solution to the differential equation.<\/p>\n

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28.\u00a0<\/strong>[latex]{y}^{\\prime }=3x+{e}^{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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29.\u00a0<\/strong>[latex]{y}^{\\prime }=\\text{ln}x+\\tan{x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=C-x+x\\text{ln}x-\\text{ln}\\left(\\cos{x}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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30.\u00a0<\/strong>[latex]{y}^{\\prime }=\\sin{x}{e}^{\\cos{x}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]{y}^{\\prime }={4}^{x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=C+\\frac{{4}^{x}}{\\text{ln}\\left(4\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]{y}^{\\prime }={\\sin}^{-1}\\left(2x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>[latex]{y}^{\\prime }=2t\\sqrt{{t}^{2}+16}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{2}{3}\\sqrt{{t}^{2}+16}\\left({t}^{2}+16\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]{x}^{\\prime }=\\text{coth}t+\\text{ln}t+3{t}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]{x}^{\\prime }=t\\sqrt{4+t}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]x=\\frac{2}{15}\\sqrt{4+t}\\left(3{t}^{2}+4t - 32\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]{y}^{\\prime }=y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>[latex]{y}^{\\prime }=\\frac{y}{x}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=Cx[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Solve the following initial-value problems starting from [latex]y\\left(t=0\\right)=1[\/latex] and [latex]y\\left(t=0\\right)=-1[\/latex]. Draw both solutions on the same graph.<\/p>\n

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38.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2t[\/latex]<\/div>\n<\/div>\n<\/div>\n
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39.\u00a0<\/strong>[latex]\\frac{dy}{dt}=\\text{-}t[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=1-\\frac{{t}^{2}}{2},y=-\\frac{{t}^{2}}{2}-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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40.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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41.\u00a0<\/strong>[latex]\\frac{dy}{dt}=\\text{-}y[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y={e}^{\\text{-}t},y=\\text{-}{e}^{\\text{-}t}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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42.\u00a0<\/strong>[latex]\\frac{dy}{dt}=2[\/latex]<\/div>\n<\/div>\n<\/div>\n

Solve the following initial-value problems starting from [latex]{y}_{0}=10[\/latex]. At what time does [latex]y[\/latex] increase to [latex]100[\/latex] or drop to [latex]1?[\/latex]<\/p>\n

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43.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4t[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=2\\left({t}^{2}+5\\right),t=3\\sqrt{5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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44.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>[latex]\\frac{dy}{dt}=-2y[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=10{e}^{-2t},t=-\\frac{1}{2}\\text{ln}\\left(\\frac{1}{10}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>[latex]\\frac{dy}{dt}={e}^{4t}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>[latex]\\frac{dy}{dt}={e}^{-4t}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{1}{4}\\left(41-{e}^{-4t}\\right)[\/latex], never<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from [latex]y\\left(t=0\\right)=-10[\/latex] to [latex]y\\left(t=0\\right)=10[\/latex] increasing by [latex]2[\/latex]. Is there some critical point where the behavior of the solution begins to change?<\/p>\n

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48. [T]<\/strong> [latex]{y}^{\\prime }=y\\left(x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n
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49. [T]<\/strong> [latex]x{y}^{\\prime }=y[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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50. [T]<\/strong> [latex]{y}^{\\prime }={t}^{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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51. [T]<\/strong> [latex]{y}^{\\prime }=x+y[\/latex] (Hint:<\/em> [latex]y=C{e}^{x}-x - 1[\/latex] is the general solution)<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
52. [T]<\/strong> [latex]{y}^{\\prime }=x\\text{ln}x+\\sin{x}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>Find the general solution to describe the velocity of a ball of mass [latex]1\\text{lb}[\/latex] that is thrown upward at a rate [latex]a[\/latex] ft\/sec.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]v\\left(t\\right)=-32t+a[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>In the preceding problem, if the initial velocity of the ball thrown into the air is [latex]a=25[\/latex] ft\/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>You throw two objects with differing masses [latex]{m}_{1}[\/latex] and [latex]{m}_{2}[\/latex] upward into the air with the same initial velocity [latex]a[\/latex] ft\/s. What is the difference in their velocity after [latex]1[\/latex] second?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]0[\/latex] ft\/s<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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56. [T]<\/strong> You throw a ball of mass [latex]1[\/latex] kilogram upward with a velocity of [latex]a=25[\/latex] m\/s on Mars, where the force of gravity is [latex]g=-3.711[\/latex] m\/s2<\/sup>. Use your calculator to approximate how much longer the ball is in the air on Mars than on Earth, where [latex]g= -9.8 m\/s^{2}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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57. [T]<\/strong> For the previous problem, use your calculator to approximate how much higher the ball went on Mars, where [latex]g= -9.8 m\/s^{2}[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]4.86[\/latex] meters<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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58. [T]<\/strong> A car on the freeway accelerates according to [latex]a=15\\cos\\left(\\pi t\\right)[\/latex], where [latex]t[\/latex] is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of [latex]51[\/latex] mph. After [latex]40[\/latex] minutes of driving, what is the driver\u2019s velocity?<\/div>\n<\/div>\n<\/div>\n
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59. [T]<\/strong> For the car in the preceding problem, find the expression for the distance the car has traveled in time [latex]t[\/latex], assuming an initial distance of [latex]0[\/latex]. How long does it take the car to travel [latex]100[\/latex] miles? Round your answer to hours and minutes.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]x=50t-\\frac{15}{{\\pi }^{2}}\\cos\\left(\\pi t\\right)+\\frac{3}{{\\pi }^{2}},2[\/latex] hours [latex]1[\/latex] minute<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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60. [T]<\/strong> For the previous problem, find the total distance traveled in the first hour.<\/div>\n<\/div>\n<\/div>\n
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61.\u00a0<\/strong>Substitute [latex]y=B{e}^{3t}[\/latex] into [latex]{y}^{\\prime }-y=8{e}^{3t}[\/latex] to find a particular solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=4{e}^{3t}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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62.\u00a0<\/strong>Substitute [latex]y=a\\cos\\left(2t\\right)+b\\sin\\left(2t\\right)[\/latex] into [latex]{y}^{\\prime }+y=4\\sin\\left(2t\\right)[\/latex] to find a particular solution.<\/div>\n<\/div>\n<\/div>\n
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63.\u00a0<\/strong>Substitute [latex]y=a+bt+c{t}^{2}[\/latex] into [latex]{y}^{\\prime }+y=1+{t}^{2}[\/latex] to find a particular solution.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=3 - 2t+{t}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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64.\u00a0<\/strong>Substitute [latex]y=a{e}^{t}\\cos{t}+b{e}^{t}\\sin{t}[\/latex] into [latex]{y}^{\\prime }=2{e}^{t}\\cos{t}[\/latex] to find a particular solution.<\/div>\n<\/div>\n<\/div>\n
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65.\u00a0<\/strong>Solve [latex]{y}^{\\prime }={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex] and solve [latex]{y}^{\\prime }=1[\/latex] with the same initial condition. As [latex]k[\/latex] approaches [latex]0[\/latex], what do you notice?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{1}{k}\\left({e}^{kt}-1\\right)[\/latex] and [latex]y=x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
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