Linear Second-Order Equations

Find a particular solution to the differential equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{9}$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{10}$

Find a particular solution to the differential equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{+}\mathbf{y}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{2}}^x$

Find a particular solution to the differential equation.

$\mathbf{2}\mathbf{x}\mathbf{\text{'}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{3}{\mathbf{t}}^2$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{sin}\mathbf{3}\mathbf{t}$

Find a particular solution to the differential equation.

$\mathbf{2}\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{9}{\mathbf{e}}^{2t}$

Find a particular solution to the differential equation.

$\frac{{\mathbf{d}}^2\mathbf{y}}{{\mathbf{dx}}^2}\mathbf{-}\mathbf{5}\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{xe}}^x$

Find a particular solution to the differential equation.

$\mathbf{\theta }\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{-}\mathbf{\theta }\left(\mathbf{t}\right)\mathbf{=}\mathbf{tsint}$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{8}\mathbf{sin}\left(\mathbf{2}\mathbf{t}\right)$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{8}{\mathbf{e}}^t$

Find a particular solution to the differential equation.$\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{11}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{te}}^{-3t}$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{16}\mathbf{tsin}\left(\mathbf{2}\mathbf{t}\right)$

Find a particular solution to the differential equation.

$\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{-}\mathbf{4}\mathbf{x}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{+}\mathbf{4}\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{te}}^{2t}$

Find a particular solution to the differential equation.

$\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{+}\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\mathbf{24}{\mathbf{t}}^2{\mathbf{e}}^t$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{=}{\mathbf{\theta }}^2$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{+}\mathbf{y}\left(\mathbf{x}\right)\mathbf{=}\mathbf{4}\mathbf{xcosx}$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{111}{\mathbf{e}}^{2t}\mathbf{cos}\mathbf{3}\mathbf{t}$

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{te}}^{-t}\mathbf{cost}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{t}}^3\mathbf{sin}\mathbf{3}\mathbf{t}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{5}{\mathbf{t}}^6{\mathbf{e}}^{3t}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{=}{\mathbf{t}}^4{\mathbf{e}}^t$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{7}{\mathbf{e}}^t\mathbf{cost}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{8}{\mathbf{t}}^{\mathbf{3}}{\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{sint}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)  $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathbf{y}\mathbf{=}\mathbf{2}{\mathbf{t}}^6{\mathbf{e}}^{-3t}$

In Problems 33, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{sint}$

In Problems 34, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}{\mathbf{e}}^{-t}$

In Problems 35, use the method of undetermined coefficients to find a particular solution to the given higher-order equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}{\mathbf{te}}^t$

In Problems 36, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. ${\mathit{y}}^{\mathbf{4}}\mathbf{-}\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{8}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$

Given that ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{cost}$  is a solution to $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{sint}$ and ${\mathbf{y}}_2\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{e}}^{2t}}{3}$ is a solution to $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{e}}^{2t}$. Use the  superposition principle to find solutions to the following differential equations: 

$\left(\mathbf{a}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{5}\mathbf{sint}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{sint}\mathbf{-}\mathbf{3}{\mathbf{e}}^{2t}$

$\left(\mathbf{c}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sint}\mathbf{+}\mathbf{18}{\mathbf{e}}^{2t}$

Given that ${\mathbf{y}}_1\left(\mathbf{t}\right)\mathbf{=}\frac{1}{4}\mathbf{sin}\mathbf{2}\mathbf{t}$  is a solution to $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{cos}\mathbf{2}\mathbf{t}$  and ${\mathbf{y}}_2\left(\mathbf{t}\right)\mathbf{=}\frac{t}{4}\mathbf{-}\frac{1}{8}$  is a solution to  $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{t}$, use the superposition principle to find solutions to the following differential equations:

$\left(\mathbf{a}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{+}\mathbf{cos}\mathbf{2}\mathbf{t}$

$\left(\mathbf{b}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{t}\mathbf{-}\mathbf{3}\mathbf{cos}\mathbf{2}\mathbf{t}$

$\left(\mathbf{c}\right)\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{11}\mathbf{t}\mathbf{-}\mathbf{12}\mathbf{cos}\mathbf{2}\mathbf{t}$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{,}{\mathbf{y}}_p\left(\mathbf{t}\right)\mathbf{=}\mathbf{-}\mathbf{t}$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathbf{y}}_p\left(\mathbf{t}\right)\mathbf{=}\mathbf{t}$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.$\mathbf{\theta }\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{\theta }\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{\theta }\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{t}\mathbf{,}{\mathbf{\theta }}_p\left(\mathbf{t}\right)\mathbf{=}\mathbf{t}\mathbf{-}\mathbf{1}$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{6}{\mathbf{x}}^2\mathbf{+}\mathbf{10}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{+}\mathbf{12}{\mathbf{e}}^x\mathbf{,}{\mathbf{y}}_p\left(\mathbf{x}\right)\mathbf{=}{\mathbf{e}}^x\mathbf{+}{\mathbf{x}}^2$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{2}{\mathbf{tan}}^3\mathbf{x}\mathbf{,}{\mathbf{y}}_p\left(\mathbf{x}\right)\mathbf{=}\mathbf{tanx}$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{+}\mathbf{2}{\mathit{e}}^{\mathit{x}}\mathbf{,} {\mathit{y}}_{\mathit{p}}\left(\mathit{x}\right)\mathbf{=}{\mathit{x}}^{\mathbf{2}}{\mathit{e}}^{\mathit{x}}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathit{y}\mathbf{=}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{t}\mathbf{-}{\mathit{t}}^{\mathbf{2}}{\mathit{e}}^{\mathit{t}}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}{\left({\mathit{e}}^{\mathit{t}}\mathbf{+}\mathit{t}\right)}^{\mathbf{2}}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{-}{\mathbf{t}}^2{\mathbf{e}}^{3t}\mathbf{+}\frac{1}{t}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}{\mathbf{e}}^t\mathbf{+}\mathbf{7}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{cosht}\mathbf{+}{\mathbf{sin}}^3\mathbf{t}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.

$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{t}\mathbf{+}{\mathbf{sin}}^2\mathbf{t}\mathbf{+}\mathbf{3}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.

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

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{t}}^2{\mathbf{e}}^{-t}\mathbf{sint}\mathbf{-}\mathbf{8}\mathbf{tcos}\mathbf{3}\mathbf{t}\mathbf{+}{\mathbf{10}}^t$

Find a general solution to the differential equation.

Find a general solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{11}\mathbf{t}\mathbf{+}\mathbf{1}$

Find a general solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{+}\mathbf{2}\mathbf{y}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{e}}^x\mathbf{sinx}$

Find a general solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{sin\theta }\mathbf{-}\mathbf{cos\theta }$

Find a general solution to the differential equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{2}\mathbf{y}\left(\mathbf{\theta }\right)\mathbf{=}{\mathbf{e}}^{-\theta }\mathbf{cos\theta }$

Find a general solution to the differential equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}\mathbf{10}\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{10}{\mathbf{x}}^4\mathbf{+}\mathbf{24}{\mathbf{x}}^3\mathbf{+}\mathbf{2}{\mathbf{x}}^2\mathbf{-}\mathbf{12}\mathbf{x}\mathbf{+}\mathbf{18}$