Exam 17: Second-Order Differential Equations

A spring with a $16$ -kg mass has natural length $0.8$ m and is maintained stretched to a length of $1.2$ m by a force of $19.6$ N. If the spring is compressed to a length of $0.4$ m and then released with zero velocity, find the position $x ( t )$ of the mass at any time $t$ .

Solve the initial value problem. $y ^ { \prime \prime } + 2 y ^ { \prime } - 3 y = 0 \quad y ( 0 ) = 7 \quad y ^ { \prime } ( 0 ) = 1$

The figure shows a pendulum with length L and the angle $\theta$ from the vertical to the pendulum. It can be shown that $\theta$ , as a function of time, satisfies the nonlinear differential equation $\frac { d ^ { 2 } \theta } { d t ^ { 2 } } + \frac { g } { L } \sin \theta = 0$ where $g = 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 }$ $\text { is the acceleration due to gravity. For small values of }$ $\theta$ we can use the linear approximation $\sin \theta = \theta$ $\text { and then the differential equation becomes linear. Find the equation }$ $\text { of motion of a pendulum with length } 1 \mathrm {~m} \text { if } \theta \text { is initially } 0.2 \mathrm { rad } \text { and the initial }$ $\text { angular velocity is }$ $\frac { d \theta } { d t } = 1 \mathrm { rad } / \mathrm { s }$

A series circuit consists of a resistor $R = 32 \Omega$ , an inductor with $L = 4 H$ , a capacitor with $C = 0.003125 \mathrm {~F}$ , and a $24$ -V battery. If the initial charge is 0.0008 C and the initial current is 0, find the current I(t) at time t.

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

Suppose a spring has mass M and spring constant k and let $\omega = \sqrt { k / M }$ . Suppose that the damping constant is so small that the damping force is negligible. If an external force $F ( t ) = 8 F _ { 0 } \cos ( \omega t )$ is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to find the equation that describes the motion of the mass.

A spring with a mass of 2 kg has damping constant 14, and a force of $3.6$ N is required to keep the spring stretched $0.3$ m beyond its natural length. The spring is stretched 1m beyond its natural length and then released with zero velocity. Find the position x(t) of the mass at any time t.

A spring with a 3-kg mass is held stretched 0.9 m beyond its natural length by a force of 30 N. If the spring begins at its equilibrium position but a push gives it an initial velocity of $1$ m/s, find the position x(t) of the mass after t seconds.

Solve the initial-value problem. $y ^ { \prime \prime } + 49 y = 0 , y \left( \frac { \pi } { 7 } \right) = 0 , y ^ { \prime } \left( \frac { \pi } { 7 } \right) = 1$

A series circuit consists of a resistor $R = 20 \Omega$ an inductor with L = $1$ H, a capacitor with C = $0.00200$ F, and a $12$ -V battery. If the initial charge and current are both 0, find the charge Q(t) at time t.

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 4 y ^ { \prime } + 3 y = 2 \sin x$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 5 y ^ { \prime } + 4 y = \sin x$

Solve the initial-value problem. $2 y ^ { \prime \prime } + 21 y ^ { \prime } + 54 y = 0 , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5$

Solve the initial-value problem. $y ^ { \prime \prime } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { \prime } \left( \frac { \pi } { 9 } \right) = 8$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = \frac { e ^ { - 2 x } } { x ^ { 3 } }$

A series circuit consists of a resistor $R = 96 \Omega$ , an inductor with $L = 8 \mathrm { H }$ , a capacitor with $C = 0.00125 \mathrm {~F}$ , and a generator producing a voltage of $E ( t ) = 48 \cos ( 10 t )$ If the initial charge is $Q = 0.001 \mathrm { C }$ and the initial current is 0, find the charge $Q ( t )$ at time t.

Solve the initial-value problem. $y ^ { \prime \prime } + 2 y ^ { \prime } - 3 y = 0 , y ( 0 ) = 2 , y ^ { \prime } ( 0 ) = 1$

Solve the differential equation. $49 y ^ { \prime \prime } + y = 0$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 9 y = 7 e ^ { 2 x }$

Solve the initial-value problem. $y ^ { \prime \prime } - 2 y ^ { \prime } - 24 y = 0 , y ( 1 ) = 4 , y ^ { \prime } ( 1 ) = 5$ .