91Ó°ÊÓ

In Problems 11–18, find a general solution to the differential equation

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{θ}\mathbf{\right)}\mathbf{+}\mathbf{y}\mathbf{\left(}\mathbf{θ}\mathbf{\right)}\mathbf{=}\mathbf{se}{\mathbf{c}}^{\mathbf{3}}\mathbf{θ}$.

Verify that the formulas for the Bessel functions${\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{t}\right),{\mathrm{Y}}_{\frac{1}{2}}\left(\mathrm{t}\right)$ do indeed solve equation (16).

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 3 N/sec. At time t = 0, an external force $2\sin \left(2\mathrm{t}+\frac{\mathrm{Ï€}}{4}\right)$N is applied to the system. Determine the amplitude and frequency of the steady-state solution.

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

In Problems 13–20, solve the given initial value problem.

${\mathit{y}}^{\mathbf{"}}\mathbf{-}\mathbf{4}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{3}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{:}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$

Find a particular solution to the differential equation.

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

Use the mass-spring oscillator analogy to decide whether all solutions to each of the following differential equations are bounded as $\mathbf{t}→\mathbf{+}\mathrm{∞}$

1. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{t}}^2\mathbf{y}\mathbf{=}\mathbf{0}$
   
2. $\mathbf{y}\mathbf{"}\mathbf{-}{\mathbf{t}}^2\mathbf{y}\mathbf{=}\mathbf{0}$
   
3. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{y}}^5\mathbf{=}\mathbf{0}$
   
4. $\mathbf{y}\mathbf{"}\mathbf{+}{\mathbf{y}}^6\mathbf{=}\mathbf{0}$
   
5. $\mathbf{y}\mathbf{"}\mathbf{+}(\mathbf{4}\mathbf{+}\mathbf{2}\mathbf{cost})\mathbf{y}\mathbf{=}\mathbf{0}$(Mathieu’s equation)
6. $\mathbf{y}\mathbf{"}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$
7. $\mathbf{y}\mathbf{"}\mathbf{-}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution for $\mathbf{t}\mathbf{<}\mathbf{0}$.

localid="1664182483957" $\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\frac{\mathbf{1}}{\mathbf{t}}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\frac{\mathbf{5}}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

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.

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

In Problems 11–18, find a general solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{sect}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$