Summary of Applications

Essential Concepts

Second-order constant-coefficient differential equations can be used to model spring-mass systems.
An examination of the forces on a spring-mass system results in a differential equation of the form $mx^{\prime\prime}+bx^{\prime}+kx=f(t)$ , where $m$ represents the mass, $b$ is the coefficient of the damping force, $k$ is the spring constant, and $f(t)$ represents any net external forces on the system.
If $b=0$ , there is no damping force acting on the system, and simple harmonic motion results. If $b\ne{0}$ , the behavior of the system depends on whether $b^{2}-4mk>0$ , $b^{2}-4mk=0$ , or $b^{2}-4mk<0$ .
If $b^2-4mk>0$ , the system is overdamped and does not exhibit oscillatory behavior.
If $b^2-4mk=0$ , the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.
If $b^2-4mk<0$ , the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.
If $f(t)\ne{0}$ , the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system.
The charge on the capacitor in an $RLC$ series circuit can also be modeled with a second-order constant-coefficient differential equation of the form $L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$ , where $L$ is the inductance, $R$ is the resistance, $C$ is the capacitance, and $E(t)$ is the voltage source.

Key Equations

Equation of simple harmonic motion
$x^{\prime\prime}+\omega^{2}x=0$
Solution for simple harmonic motion
$x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}$
Alternative form of solution for SHM
$x(t)=A\sin{(\omega{t}+\phi)}$
Forced harmonic motion
$mx^{\prime\prime}+bx^{\prime}+kx=f(t)$

Charge in a RLC series circuit
$L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$

Glossary

RLC series circuit: a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit

simple harmonic motion: motion described by the equation $x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}$ as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely

steady-state solution: a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution

Module 7: Second-Order Differential Equations

Summary of Applications

Essential Concepts

Key Equations

Glossary

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