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Find a general solution for $\mathbf{t}\mathbf{<}\mathbf{0}$.${\mathbf{t}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{9}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{17}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Find a general solution to the differential equation.

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

$\mathit{y}\mathbf{"}\mathbf{-}\mathbf{6}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{:}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{25}}{\mathbf{3}}$

Find a particular solution to the differential equation.

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

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

$\frac{\mathbf{1}}{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{tan}\mathbf{2}\mathbf{t}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{e}}^{\mathbf{t}}$

Armageddon. Earth revolves around the sun in an approximately circular orbit with radius r = a, completing a revolution in the time $\mathrm{T}=2\mathrm{蟺}{\left(\frac{{\mathrm{a}}^3}{\mathrm{GM}}\right)}^{\frac{1}{2}}$, which is one Earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on Earth is given by $\frac{\mathrm{GMm}}{{\mathrm{r}}^2}$, where m is the mass of Earth. Therefore, if Earth 鈥渟tood still,鈥� losing its orbital velocity, it would fall on a straight line into the sun in accordance with Newton鈥檚 second law: $\mathrm{m}\frac{{\mathrm{d}}^2\mathrm{r}}{{\mathrm{dt}}^2}=-\frac{\mathrm{GMm}}{{\mathrm{r}}^2}$.

If this calamity occurred, what fraction of the normal year T would it take for Earth to splash into the sun (i.e., achieve r = 0)? [Hint: Use the energy integral lemma and the initial conditions$\mathrm{r}\left(0\right)=\mathrm{a},\mathrm{r}\text{'}\left(0\right)=0$ .]

Consider the equation for free mechanical vibration,$\mathrm{my}+\mathrm{by}\text{'}+\mathrm{ky}=0$ , and assume the motion is critically damped. Let $\mathrm{y}\left(0\right)={\mathrm{y}}_0,\mathrm{y}\text{'}\left(0\right)={\mathrm{v}}_0$and assume ${\mathrm{y}}_0鈮�0$.

1. Prove that the mass will pass through its equilibrium at exactly one positive time if and only if $\frac{-2{\mathrm{my}}_0}{2{\mathrm{mv}}_0+{\mathrm{by}}_0}>0$.
2. Use computer software to illustrate part (a) for a specific choice of m, b, k,${\mathrm{y}}_0$ , and ${\mathrm{v}}_0$. Be sure to include an appropriate graph in your illustration.

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 differential equation.

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

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

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{3}}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}$