{"id":94,"date":"2021-03-25T02:20:59","date_gmt":"2021-03-25T02:20:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/first-order-linear-equations-2\/"},"modified":"2021-11-17T02:54:24","modified_gmt":"2021-11-17T02:54:24","slug":"first-order-linear-equations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/first-order-linear-equations-2\/","title":{"raw":"Problem Set: First-order Linear Equations","rendered":"Problem Set: First-order Linear Equations"},"content":{"raw":"

Are the following differential equations linear? Explain your reasoning.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]\\frac{dy}{dx}={x}^{2}y+\\sin{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>[latex]\\frac{dy}{dt}=ty[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"198576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"198576\"]Yes[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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

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[reveal-answer q=\"489753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489753\"]Yes[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]y^{\\prime} =y+{e}^{y}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Write the following first-order differential equations in standard form.<\/p>\r\n\r\n

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

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[reveal-answer q=\"753836\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"753836\"][latex]y^{\\prime} -{x}^{3}y=\\sin{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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7.\u00a0<\/strong>[latex]y^{\\prime} +3y-\\text{ln}x=0[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>[latex]\\text{-}xy^{\\prime} =\\left(3x+2\\right)y+x{e}^{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"522748\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"522748\"][latex]y^{\\prime} +\\frac{\\left(3x+2\\right)}{x}y=\\text{-}{e}^{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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9.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4y+ty+\\tan{t}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]\\frac{dy}{dt}=yx\\left(x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"795179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"795179\"][latex]\\frac{dy}{dt}-yx\\left(x+1\\right)=0[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

What are the integrating factors for the following differential equations?<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]y^{\\prime} =xy+3[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]y^{\\prime} +{e}^{x}y=\\sin{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"311812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"311812\"][latex]{e}^{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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13.\u00a0<\/strong>[latex]y^{\\prime} =x\\text{ln}\\left(x\\right)y+3x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]\\frac{dy}{dx}=\\text{tanh}\\left(x\\right)y+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"46763\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"46763\"][latex]\\text{-}\\text{ln}\\left(\\text{cosh}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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15.\u00a0<\/strong>[latex]\\frac{dy}{dt}+3ty={e}^{t}y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the following differential equations by using integrating factors.<\/p>\r\n\r\n

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

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[reveal-answer q=\"474368\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474368\"][latex]y=C{e}^{3x}-\\frac{2}{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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17.\u00a0<\/strong>[latex]y^{\\prime} =2y-{x}^{2}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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18.\u00a0<\/strong>[latex]xy^{\\prime} =3y - 6{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"373723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"373723\"][latex]y=C{x}^{3}+6{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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19.\u00a0<\/strong>[latex]\\left(x+2\\right)y^{\\prime} =3x+y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>[latex]y^{\\prime} =3x+xy[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"121029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"121029\"][latex]y=C{e}^{\\frac{{x}^{2}}{2}}-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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21.\u00a0<\/strong>[latex]xy^{\\prime} =x+y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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22.\u00a0<\/strong>[latex]\\sin\\left(x\\right)y^{\\prime} =y+2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"390735\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390735\"][latex]y=C\\tan\\left(\\frac{x}{2}\\right)-2x+4\\tan\\left(\\frac{x}{2}\\right)\\text{ln}\\left(\\sin\\left(\\frac{x}{2}\\right)\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

23.\u00a0<\/strong>[latex]y^{\\prime} =y+{e}^{x}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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24.\u00a0<\/strong>[latex]xy^{\\prime} =3y+{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"216191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"216191\"][latex]y=C{x}^{3}-{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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25.\u00a0<\/strong>[latex]y^{\\prime} +\\text{ln}x=\\frac{y}{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?<\/p>\r\n\r\n

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26. [T]<\/strong> [latex]\\left(x+2\\right)y^{\\prime} =2y - 1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"814422\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814422\"][latex]y=C{\\left(x+2\\right)}^{2}+\\frac{1}{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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27. [T]<\/strong> [latex]y^{\\prime} =3{e}^{\\frac{t}{3}}-2y[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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28. [T]<\/strong> [latex]xy^{\\prime} +\\frac{y}{2}=\\sin\\left(3t\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"176472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176472\"][latex]y=\\frac{C}{\\sqrt{x}}+2\\sin\\left(3t\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

29. [T]<\/strong> [latex]xy^{\\prime} =2\\frac{\\cos{x}}{x}-3y[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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30. [T]<\/strong> [latex]\\left(x+1\\right)y^{\\prime} =3y+{x}^{2}+2x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"63576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"63576\"][latex]y=C{\\left(x+1\\right)}^{3}-{x}^{2}-2x - 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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31. [T]<\/strong> [latex]\\sin\\left(x\\right)y^{\\prime} +\\cos\\left(x\\right)y=2x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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32. [T]<\/strong> [latex]\\sqrt{{x}^{2}+1}y^{\\prime} =y+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"426705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426705\"][latex]y=C{e}^{{\\text{sinh}}^{-1}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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33. [T]<\/strong> [latex]{x}^{3}y^{\\prime} +2{x}^{2}y=x+1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the following initial-value problems by using integrating factors.<\/p>\r\n\r\n

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34,\u00a0<\/strong>[latex]y^{\\prime} +y=x,y\\left(0\\right)=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"865414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"865414\"][latex]y=x+4{e}^{x}-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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35.\u00a0<\/strong>[latex]y^{\\prime} =y+2{x}^{2},y\\left(0\\right)=0[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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36.\u00a0<\/strong>[latex]xy^{\\prime} =y - 3{x}^{3},y\\left(1\\right)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"781847\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"781847\"][latex]y=-\\frac{3x}{2}\\left({x}^{2}-1\\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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37.\u00a0<\/strong>[latex]{x}^{2}y^{\\prime} =xy-\\text{ln}x,y\\left(1\\right)=1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38.\u00a0<\/strong>[latex]\\left(1+{x}^{2}\\right)y^{\\prime} =y - 1,y\\left(0\\right)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"194748\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"194748\"][latex]y=1-{e}^{{\\tan}^{-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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39.\u00a0<\/strong>[latex]xy^{\\prime} =y+2x\\text{ln}x,y\\left(1\\right)=5[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40.\u00a0<\/strong>[latex]\\left(2+x\\right)y^{\\prime} =y+2+x,y\\left(0\\right)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"975624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"975624\"][latex]y=\\left(x+2\\right)\\text{ln}\\left(\\frac{x+2}{2}\\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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41.\u00a0<\/strong>[latex]y^{\\prime} =xy+2x{e}^{x},y\\left(0\\right)=2[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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42.\u00a0<\/strong>[latex]\\sqrt{x}y^{\\prime} =y+2x,y\\left(0\\right)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"597154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"597154\"][latex]y=2{e}^{2\\sqrt{x}}-2x - 2\\sqrt{x}-1[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

43.\u00a0<\/strong>[latex]y^{\\prime} =2y+x{e}^{x},y\\left(0\\right)=-1[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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44.\u00a0<\/strong>A falling object of mass [latex]m[\/latex] can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity given an initial velocity of [latex]0[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"223898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223898\"][latex]v\\left(t\\right)=\\frac{gm}{k}\\left(1-{e}^{\\frac{\\text{-}kt}{m}}\\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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45.\u00a0<\/strong>Using your expression from the preceding problem, what is the terminal velocity? (Hint:<\/em> Examine the limiting behavior; does the velocity approach a value?)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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46. [T]<\/strong> Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8[\/latex] m\/s2<\/sup> and the proportionality constant is [latex]4?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"738361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"738361\"][latex]40.451[\/latex] seconds[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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47.\u00a0<\/strong>A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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48.\u00a0<\/strong>Using your expression from the preceding problem, what is the terminal velocity? (Hint:<\/em> Examine the limiting behavior: Does the velocity approach a value?)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"682502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"682502\"][latex]\\sqrt{\\frac{gm}{k}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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49. [T]<\/strong> Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8{\\text{m\/s}}^{2}[\/latex] and the proportionality constant is [latex]4?[\/latex] Does it take more or less time than your initial estimate?<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following problems, determine how parameter [latex]a[\/latex] affects the solution.<\/p>\r\n\r\n

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50.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ax+y[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"734672\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734672\"][latex]y=C{e}^{x}-a\\left(x+1\\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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51.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ay+x[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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52.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ax+xy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"886353\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"886353\"][latex]y=C{e}^{\\frac{{x}^{2}}{2}}-a[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

53.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =x+axy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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54. <\/strong>Solve [latex]y^{\\prime} -y={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex]. As [latex]k[\/latex] approaches [latex]1[\/latex], what happens to your formula?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"24549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"24549\"][latex]y=\\frac{{e}^{kt}-{e}^{t}}{k - 1}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Are the following differential equations linear? Explain your reasoning.<\/p>\n

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1.\u00a0<\/strong>[latex]\\frac{dy}{dx}={x}^{2}y+\\sin{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]\\frac{dy}{dt}=ty[\/latex]<\/p>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
Yes<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]y^{\\prime} =y+{e}^{y}[\/latex]<\/div>\n<\/div>\n<\/div>\n

Write the following first-order differential equations in standard form.<\/p>\n

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

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[latex]y^{\\prime} -{x}^{3}y=\\sin{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]y^{\\prime} +3y-\\text{ln}x=0[\/latex]<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\text{-}xy^{\\prime} =\\left(3x+2\\right)y+x{e}^{x}[\/latex]<\/p>\n<\/div>\n

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[latex]y^{\\prime} +\\frac{\\left(3x+2\\right)}{x}y=\\text{-}{e}^{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>[latex]\\frac{dy}{dt}=4y+ty+\\tan{t}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]\\frac{dy}{dt}=yx\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{dy}{dt}-yx\\left(x+1\\right)=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

What are the integrating factors for the following differential equations?<\/p>\n

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11.\u00a0<\/strong>[latex]y^{\\prime} =xy+3[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]y^{\\prime} +{e}^{x}y=\\sin{x}[\/latex]<\/p>\n<\/div>\n

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[latex]{e}^{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]y^{\\prime} =x\\text{ln}\\left(x\\right)y+3x[\/latex]<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]\\frac{dy}{dx}=\\text{tanh}\\left(x\\right)y+1[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{-}\\text{ln}\\left(\\text{cosh}x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\frac{dy}{dt}+3ty={e}^{t}y[\/latex]<\/div>\n<\/div>\n<\/div>\n

Solve the following differential equations by using integrating factors.<\/p>\n

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

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[latex]y=C{e}^{3x}-\\frac{2}{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]y^{\\prime} =2y-{x}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>[latex]xy^{\\prime} =3y - 6{x}^{2}[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{x}^{3}+6{x}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>[latex]\\left(x+2\\right)y^{\\prime} =3x+y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]y^{\\prime} =3x+xy[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{e}^{\\frac{{x}^{2}}{2}}-3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>[latex]xy^{\\prime} =x+y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]\\sin\\left(x\\right)y^{\\prime} =y+2x[\/latex]<\/p>\n<\/div>\n

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[latex]y=C\\tan\\left(\\frac{x}{2}\\right)-2x+4\\tan\\left(\\frac{x}{2}\\right)\\text{ln}\\left(\\sin\\left(\\frac{x}{2}\\right)\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
23.\u00a0<\/strong>[latex]y^{\\prime} =y+{e}^{x}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]xy^{\\prime} =3y+{x}^{2}[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{x}^{3}-{x}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>[latex]y^{\\prime} +\\text{ln}x=\\frac{y}{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?<\/p>\n

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26. [T]<\/strong> [latex]\\left(x+2\\right)y^{\\prime} =2y - 1[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{\\left(x+2\\right)}^{2}+\\frac{1}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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27. [T]<\/strong> [latex]y^{\\prime} =3{e}^{\\frac{t}{3}}-2y[\/latex]<\/div>\n<\/div>\n<\/div>\n
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28. [T]<\/strong> [latex]xy^{\\prime} +\\frac{y}{2}=\\sin\\left(3t\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]y=\\frac{C}{\\sqrt{x}}+2\\sin\\left(3t\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
29. [T]<\/strong> [latex]xy^{\\prime} =2\\frac{\\cos{x}}{x}-3y[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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30. [T]<\/strong> [latex]\\left(x+1\\right)y^{\\prime} =3y+{x}^{2}+2x+1[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{\\left(x+1\\right)}^{3}-{x}^{2}-2x - 1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31. [T]<\/strong> [latex]\\sin\\left(x\\right)y^{\\prime} +\\cos\\left(x\\right)y=2x[\/latex]<\/div>\n<\/div>\n<\/div>\n
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32. [T]<\/strong> [latex]\\sqrt{{x}^{2}+1}y^{\\prime} =y+2[\/latex]<\/p>\n<\/div>\n

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[latex]y=C{e}^{{\\text{sinh}}^{-1}x}-2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33. [T]<\/strong> [latex]{x}^{3}y^{\\prime} +2{x}^{2}y=x+1[\/latex]<\/div>\n<\/div>\n<\/div>\n

Solve the following initial-value problems by using integrating factors.<\/p>\n

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34,\u00a0<\/strong>[latex]y^{\\prime} +y=x,y\\left(0\\right)=3[\/latex]<\/p>\n<\/div>\n

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[latex]y=x+4{e}^{x}-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]y^{\\prime} =y+2{x}^{2},y\\left(0\\right)=0[\/latex]<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]xy^{\\prime} =y - 3{x}^{3},y\\left(1\\right)=0[\/latex]<\/p>\n<\/div>\n

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[latex]y=-\\frac{3x}{2}\\left({x}^{2}-1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>[latex]{x}^{2}y^{\\prime} =xy-\\text{ln}x,y\\left(1\\right)=1[\/latex]<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>[latex]\\left(1+{x}^{2}\\right)y^{\\prime} =y - 1,y\\left(0\\right)=0[\/latex]<\/p>\n<\/div>\n

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[latex]y=1-{e}^{{\\tan}^{-1}x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39.\u00a0<\/strong>[latex]xy^{\\prime} =y+2x\\text{ln}x,y\\left(1\\right)=5[\/latex]<\/div>\n<\/div>\n<\/div>\n
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40.\u00a0<\/strong>[latex]\\left(2+x\\right)y^{\\prime} =y+2+x,y\\left(0\\right)=0[\/latex]<\/p>\n<\/div>\n

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[latex]y=\\left(x+2\\right)\\text{ln}\\left(\\frac{x+2}{2}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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41.\u00a0<\/strong>[latex]y^{\\prime} =xy+2x{e}^{x},y\\left(0\\right)=2[\/latex]<\/div>\n<\/div>\n<\/div>\n
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42.\u00a0<\/strong>[latex]\\sqrt{x}y^{\\prime} =y+2x,y\\left(0\\right)=1[\/latex]<\/p>\n<\/div>\n

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[latex]y=2{e}^{2\\sqrt{x}}-2x - 2\\sqrt{x}-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
43.\u00a0<\/strong>[latex]y^{\\prime} =2y+x{e}^{x},y\\left(0\\right)=-1[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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44.\u00a0<\/strong>A falling object of mass [latex]m[\/latex] can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity given an initial velocity of [latex]0[\/latex].<\/p>\n<\/div>\n

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[latex]v\\left(t\\right)=\\frac{gm}{k}\\left(1-{e}^{\\frac{\\text{-}kt}{m}}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>Using your expression from the preceding problem, what is the terminal velocity? (Hint:<\/em> Examine the limiting behavior; does the velocity approach a value?)<\/div>\n<\/div>\n<\/div>\n
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46. [T]<\/strong> Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8[\/latex] m\/s2<\/sup> and the proportionality constant is [latex]4?[\/latex]<\/p>\n<\/div>\n

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[latex]40.451[\/latex] seconds<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant [latex]k[\/latex]. Set up the differential equation and solve for the velocity.<\/div>\n<\/div>\n<\/div>\n
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48.\u00a0<\/strong>Using your expression from the preceding problem, what is the terminal velocity? (Hint:<\/em> Examine the limiting behavior: Does the velocity approach a value?)<\/p>\n<\/div>\n

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[latex]\\sqrt{\\frac{gm}{k}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49. [T]<\/strong> Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall [latex]5000[\/latex] meters if the mass is [latex]100[\/latex] kilograms, the acceleration due to gravity is [latex]9.8{\\text{m\/s}}^{2}[\/latex] and the proportionality constant is [latex]4?[\/latex] Does it take more or less time than your initial estimate?<\/div>\n<\/div>\n<\/div>\n

For the following problems, determine how parameter [latex]a[\/latex] affects the solution.<\/p>\n

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50.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ax+y[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/p>\n<\/div>\n

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[latex]y=C{e}^{x}-a\\left(x+1\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ay+x[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/div>\n<\/div>\n<\/div>\n
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52.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =ax+xy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/p>\n<\/div>\n

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[latex]y=C{e}^{\\frac{{x}^{2}}{2}}-a[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
53.\u00a0<\/strong>Solve the generic equation [latex]y^{\\prime} =x+axy[\/latex]. How does varying [latex]a[\/latex] change the behavior?<\/span><\/div>\n<\/div>\n<\/div>\n
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54. <\/strong>Solve [latex]y^{\\prime} -y={e}^{kt}[\/latex] with the initial condition [latex]y\\left(0\\right)=0[\/latex]. As [latex]k[\/latex] approaches [latex]1[\/latex], what happens to your formula?<\/p>\n<\/div>\n

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[latex]y=\\frac{{e}^{kt}-{e}^{t}}{k - 1}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
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