Linear Second-Order Equations

Find a general solution to the given differential equation.$\mathbf{36}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{24}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.$\mathbf{25}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.$\mathbf{16}\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{56}\mathbf{z}\mathbf{\text{'}}\mathbf{+}\mathbf{49}\mathbf{z}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.$\mathbf{u}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{11}\mathbf{u}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation. ${\mathbf{t}}^{\mathbf{2}}\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{+}\mathbf{5}\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}$

Question: Find a general solution to the given differential equation. $\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{16}\mathbf{y}\mathbf{=}{\mathbf{te}}^t$

Question: Find a general solution to the given differential equation. $\mathbf{\nu }\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{v}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{v}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation. $\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{11}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.

${\mathbf{y}}^{\mathbf{4}}\mathbf{=}\mathbf{120}\mathbf{t}$

Find a general solution to the given differential equation.

$\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{11}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution to the given differential equation.

$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{tsint}$

Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{=}\mathbf{7}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}{\mathbf{e}}^{\mathbf{t}}$

Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{33}\mathbf{y}\mathbf{=}\mathbf{546}\mathbf{sint}$

Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{16}\mathbf{y}\left(\mathbf{\theta }\right)\mathbf{=}\mathbf{tan}\left(\mathbf{4}\mathbf{\theta }\right)$

Find a general solution to the given differential equation.

${\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{xy}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{6}{\mathbf{x}}^{\mathbf{-}\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{x}\mathbf{,} \mathbf{x}\mathbf{>}\mathbf{0}$

Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{5}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{13}{\mathbf{x}}^{\mathbf{-}\mathbf{2}}\mathbf{y}\mathbf{,} \mathbf{x}\mathbf{>}\mathbf{0}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{2}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{y}\left(\mathbf{\theta }\right)\mathbf{=}\mathbf{2}\mathbf{cos\theta }\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{3}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{=}\mathbf{6}\mathbf{cos}\left(\mathbf{3}\mathbf{t}\right)\mathbf{-}\mathbf{sin}\left(\mathbf{3}\mathbf{t}\right)\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{2}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{8}$

Find the solution to the given initial value problem.

$\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\frac{\mathbf{11}}{\mathbf{2}}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{27}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{40}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{6}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{-}\mathbf{12}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{14}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\left(0\right)\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{1}$

Find the solution to the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\theta \right)\mathbf{+}\mathbf{y}\left(\theta \right)\mathbf{=}\mathbf{sec\theta }\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{2}$

Find the solution to the given initial value problem.

$\mathbf{9}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{12}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{3}$

Use the mass–spring oscillator analogy to decide whether all solutions to each of the following differential equations are bounded as \({\bf{t}} \to {\bf{ + }}\infty \)

\(\begin{array}{l}\left( {\bf{a}} \right)\,\,{\bf{y'' + }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\\\left( {\bf{b}} \right)\,\,{\bf{y'' - }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\\\left( {\bf{c}} \right)\,\,{\bf{y'' + }}{{\bf{y}}^{\bf{7}}}{\bf{ = 0}}\\\left( {\bf{d}} \right)\,\,{\bf{y'' + }}{{\bf{y}}^{\bf{8}}}{\bf{ = 0}}\\\left( {\bf{e}} \right)\,\,{\bf{y'' + }}\left( {{\bf{3 + sint}}} \right){\bf{y = 0}}\\\left( {\bf{f}} \right)\,\,{\bf{y'' + }}{{\bf{t}}^{\bf{2}}}{\bf{y' + y = 0}}\\\left( {\bf{g}} \right)\,\,{\bf{y'' - }}{{\bf{t}}^{\bf{2}}}{\bf{y' - y = 0}}\end{array}\)

A 3-kg mass is attached to a spring with stiffness k = 75 N/m, as in Figure 4.1, page 152. The mass is displaced $\frac{\mathbf{1}}{\mathbf{4}}\mathbf{m}$ to the left and given a velocity of 1 m/sec to the right. The damping force is negligible. Find the equation of motion of the mass along with the amplitude, period, and frequency. How long after release does the mass pass through the equilibrium position?

A 32-lb weight is attached to a vertical spring, causing it to stretch 6 in. upon coming to rest at equilibrium. The damping constant for the system is 2 lb-sec/ft. An external force $\mathbf{f}\left(t\right)\mathbf{=}\mathbf{4}\mathbf{cos}\mathbf{8}\mathbf{t}\mathbf{ }\mathbf{lb}$ is applied to the weight. Find the steady-state solution for the system. What is its resonant frequency?