91Ó°ÊÓ

Find a general solution to the given differential equation.

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

In Problems 1 through 4, use Theorem 5 to discuss the existence and uniqueness of a solution to the differential equation that satisfies the initial conditions $\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}{\mathbf{Y}}_{\mathbf{o}}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}{\mathbf{Y}}_{\mathbf{1}}$, where ${\mathbf{Y}}_{\mathbf{o}}$and ${\mathbf{Y}}_{\mathbf{1}}$are real constants $\mathbf{(}\mathbf{1}\mathbf{+}{\mathbf{t}}^{\mathbf{2}}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{tant}$.

Question: Find a general solution to the given differential equation.

$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}$

Verify that for $\mathit{b}\mathbf{=}\mathbf{0}$and ${\mathit{F}}_{\mathit{e}\mathit{x}\mathit{t}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$, equation (3) has a solution of the form $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{³¦´Ç²õÓ\neg ³Ù}\mathbf{,}\mathbf{Ó\neg }\mathbf{=}\sqrt{\mathbf{k}\mathbf{/}\mathbf{m}}$

In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{tan}\mathbf{2}\mathbf{t}$

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 role="math" localid="1654926813168" $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{e}}^{2t}$. Use the superposition principle to find solutions to the following differential equations:

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

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

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

Show that if y(t) satisfies $\mathbf{y}\mathbf{"}\mathbf{-}\mathbf{ty}\mathbf{=}\mathbf{0}$, then y(-t) satisfies $\mathbf{y}\mathbf{"}\mathbf{+}\mathbf{ty}\mathbf{=}\mathbf{0}$.

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}{\mathbf{t}}^{-1}{\mathbf{e}}^t$

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System and role="math" localid="1663948416136" $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$for the MKS system.

Sketch the frequency response curve (13) for the system in which m = 4, k = 1, b = 2.

Find a general solution to the given differential equation$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{0}$.