Exam 17: Second-Order Differential Equations

A series circuit consists of a resistor $R = 20 \Omega$ , an inductor with $L = 1 \mathrm { H }$ , a capacitor with $C = 0.00200 \mathrm {~F}$ , and a $12 - \mathrm { V }$ 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. $y ^ { tt } - 5 y ^ { t } + 4 y = 4 \sin x$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 3 y ^ { t } = e ^ { 9 x }$

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

Solve the differential equation using the method of variation of parameters. $y ^ { tt } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }$

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$ .

A spring with a mass of $2 \mathrm {~kg}$ has damping constant 8 and spring constant 80 . Graph the position function of the mass at time $t$ if it starts at the equilibrium position with a velocity of $2 \mathrm {~m} / \mathrm { s }$ .

Solve the differential equation. $y ^ {tt } - 2 y ^ { t} - 15 y = 0$

Use power series to solve the differential equation. $y ^ { tt } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ {t } ( 0 ) = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } + 25 y = x$

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

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } - 4 y ^ { t } + 5 y = 8 e ^ { - x }$

Solve the boundary-value problem, if possible. $y ^ { tt } + 5 y ^ { t } - 14 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

Find a trial solution for the method of undetermined coefficients.Do not determine the coefficients. $y ^ { tt} + 3 y ^ { t } + 5 y = x ^ { 4 } e ^ { 5 x }$

Solve the initial-value problem. $y ^ { tt } - 2 y ^ { t } - 24 y = 0 , y ( 1 ) = 4 , y ^ { t } ( 1 ) = 6$

A spring with a mass of $9 \mathrm {~kg}$ has damping constant 28 and spring constant 192 . Find the damping constant that would produce critical damping.

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt} + 6 y ^ { t } + 9 y = 2 + x$

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 $F ( t ) = 4 F _ { 0 } \cos ( \omega t )$ 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 $2 \mathrm {~kg}$ has damping constant 14 , and a force of $4.8 \mathrm {~N}$ is required to keep the spring stretched $0.4 \mathrm {~m}$ beyond its natural length. Find the mass that would produce critical damping.

Solve the differential equation. $36 y ^ {tt } + y = 0$