Exam 17: Second-Order Differential Equations

A spring with a mass of $9 \mathrm {~kg}$ has damping constant 28 and spring constant 195 . Find the damping constant that would produce critical damping. Select the correct answer.

Solve the differential equation. $y ^ { tt } - 8 y ^ { t } + 32 y = 0$

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

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { tt } + 2 y ^ { t } = 1 + x e ^ { - 4 x }$

Solve the differential equation. $y ^ { tt } - 2 y ^ { t } = e ^ { 3 x }$

Solve the differential equation using the method of variation of parameters. Select the correct answer. $y ^ { tt } - y ^ { t } = e ^ { 2 x }$

Use power series to solve the differential equation. $y ^ { t} = 4 x y$

Solve the initial-value problem. $y ^ { tt } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ {t } \left( \frac { \pi } { 9 } \right) = 4$

Solve the initial-value problem. $y ^ { tt } + 81 y = 0 , y \left( \frac { \pi } { 9 } \right) = 0 , y ^ { t } \left( \frac { \pi } { 9 } \right) = 6$

The solution of the initial-value problem $x ^ { 2 } y ^ { tt } + x y ^ { t } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.

Solve the differential equation using the method of variation of parameters. $y ^ {tt } + 10 y ^ { t} + 25 y = \frac { e ^ { - 5 x } } { x ^ { 3 } }$

Solve the differential equation using the method of variation of parameters. $y ^ { tt } - 9 y ^ { t } = \frac { 1 } { x }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { tt } + 5 y ^ {t } + 6 y = x ^ { 2 }$

Solve the initial-value problem. $y ^ { tt } + 8 y ^ {t } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 5$

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

Solve the initial-value problem. $y ^ { tt } + 3 y ^ { t } - 4 y = 0 , y ( 0 ) = 2 , y ^ { t } ( 0 ) = 1$

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.

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

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.

The solution of the initial-value problem $x ^ { 2 } y ^ {tt } + x y ^ { t } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.