Linear Second-Order Equations

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.

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

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.

${\mathbf{e}}^{\mathbf{t}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{t}\mathbf{-}\mathbf{3}}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{lnt}$

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.

${\mathbf{t}}^{\mathbf{2}}\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{tz}\mathbf{\text{'}}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{cost}\mathbf{;}\mathbf{z}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{z}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{yy}\mathbf{\text{'}}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.

$\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{t}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{sint}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.

${\mathbf{t}}^2\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{ty}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.${\mathbf{t}}^2\frac{{\mathbf{d}}^2\mathbf{y}}{{\mathbf{dt}}^2}\mathbf{+}\mathbf{2}\mathbf{t}\frac{\mathrm{dy}}{\mathrm{dt}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{0}$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.

${\mathbf{t}}^2\frac{{\mathbf{d}}^2\mathbf{z}}{{\mathbf{dt}}^2}\mathbf{+}\mathbf{5}\mathbf{t}\frac{\mathrm{dz}}{\mathrm{dt}}\mathbf{+}\mathbf{4}\mathbf{z}\mathbf{=}\mathbf{0}$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.

$\mathbf{1}2.\frac{{\mathbf{d}}^2\mathbf{w}}{{\mathbf{dt}}^2}\mathbf{+}\frac{6}{t}\frac{\mathrm{dw}}{\mathrm{dt}}\mathbf{+}\frac{4}{{\mathbf{t}}^2}\mathbf{w}\mathbf{=}\mathbf{0}$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.$\mathbf{13}\mathbf{.}\mathbf{9}{\mathbf{t}}^2\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{15}\mathbf{ty}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Find a general solution to the given Cauchy-Euler equation for $\mathbf{t}\mathbf{>}\mathbf{0}$

${\mathbf{t}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

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

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

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

${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

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 for  $\mathbf{t}\mathbf{<}\mathbf{0}$.${\mathbf{t}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{3}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Solve the given initial value problem for the Cauchy-Euler equation.

${\mathbf{t}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{4}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{11}$

Solve the given initial value problem for the Cauchy-Euler equation.$\begin{array}{l}{\mathbf{t}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{ty}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{;}\\ \mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{13}\end{array}$

Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation. ${\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{)}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{7}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{>}\mathbf{2}$

Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation.${\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{)}}^2\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{10}\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{14}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{>}\mathbf{-}\mathbf{1}$

Let ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ be two functions defined on $\mathbf{(}\mathbf{-}\infty \mathbf{,}\infty \mathbf{)}$.

1. True or False: If ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the interval $\left[a,b\right]$, then ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the smaller interval $\mathbf{[}\mathbf{c}\mathbf{,}\mathbf{d}\mathbf{]}\subset \mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$.
2. True or False: If ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the interval $\left[\mathbf{a}\mathbf{,}\mathbf{b}\right]$, then ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly dependent on the larger interval $\mathbf{[}\mathbf{C}\mathbf{,}\mathbf{D}\mathbf{]}\supset \mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$.

Let ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$ and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left|{\mathbf{t}}^{\mathbf{3}}\right|$. Are ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ linearly independent on the following intervals? 

(a). $\mathbf{[}\mathbf{0}\mathbf{,}\infty \mathbf{)}$

(b). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}$

(c). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

(d) Compute the Wronskian $\mathbf{W}\left[{\mathbf{y}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\right]\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ on the interval $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Consider the linear equation ${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\text{'}\text{'}\mathbf{-}\mathbf{3}\mathbf{ty}\text{'}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$ for $\mathbf{-}\mathbf{\infty }\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{\infty }$

(a). Verify that ${\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{=}\mathbf{t}$ and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$ are two solutions to 21 on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$. Furthermore, show that ${\mathbf{y}}_{\mathbf{1}}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{y}}_{\mathbf{2}}^{}\text{'}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{-}\mathbf{y}}_{\mathbf{1}}^{}\text{'}\left({\mathbf{t}}_{\mathbf{0}}\right){\mathbf{y}}_{\mathbf{2}}\left({\mathbf{t}}_{\mathbf{0}}\right)\ne \mathbf{0}\mathbf{,}{\mathbf{t}}_{\mathbf{0}}\mathbf{=}\mathbf{1}$.

(b). Prove that ${\mathbf{y}}_1\left(\mathbf{t}\right)$ and ${\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$ are linearly independent on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

(c). Verify that the function ${\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{|}\mathbf{t}{\mathbf{|}}^{\mathbf{3}}$ is also a solution to 21 on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

(d). Prove that there is no choice of constants ${\mathbf{c}}_{\mathbf{1}}\mathbf{,}{\mathbf{c}}_{\mathbf{2}}$ such that ${\mathbf{y}}_{\mathbf{3}}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$ for all t in $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$. [Hint: Argue that the contrary assumption leads to a contradiction.]

(e). From parts (c) and (d), we see that there is at least one solution to 21 on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$ that is not expressible as a linear combination of the solutions ${\mathbf{y}}_{\mathbf{1}}\left(\mathbf{t}\right)\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)$. Does this provide a counterexample to the theory in this section? Explain.

Let ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}$ and ${\mathbf{y}}_{\mathbf{2}}\left(\mathbf{t}\right)\mathbf{=}\mathbf{2}\mathbf{t}\left|\mathbf{t}\right|$. Are ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_2$ linearly independent on the interval

(a). $\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$

(b). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{0}\mathbf{]}$

(c). $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

(d). Compute the Wronskian $\mathbf{W}\left[{\mathbf{y}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{2}}\right]\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ on the interval $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Prove that if ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly independent solutions of  $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{py}\mathbf{\text{'}}\mathbf{+}\mathbf{qy}\mathbf{=}\mathbf{0}$ on $(\mathbf{a}\mathbf{,}\mathbf{b})$, then they cannot both be zero at the same point  ${\mathbf{t}}_{\mathbf{0}}$ in $\left(a,b\right)$

Superposition Principle. Let ${\mathbf{y}}_{\mathbf{1}}$ be a solution to $\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{g}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ on the interval I and let ${\mathbf{y}}_{\mathbf{2}}$ be a solution to $\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{g}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ on the same interval. Show that for any constants ${\mathbf{k}}_{\mathbf{1}}$ and ${\mathbf{k}}_{\mathbf{2}}$, the function ${\mathbf{k}}_{\mathbf{1}}{\mathbf{y}}_{\mathbf{1}}\mathbf{+}{\mathbf{k}}_{\mathbf{2}}{\mathbf{y}}_{\mathbf{2}}$ is a solution on I to $\mathbf{y}\text{'}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\text{'}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{k}}_{\mathbf{1}}{\mathbf{g}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}{\mathbf{k}}_{\mathbf{2}}{\mathbf{g}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$.

Determine whether the following functions can be Wronskians on $\mathbf{-}\mathbf{1}\mathbf{<}\mathbf{t}\mathbf{<}\mathbf{1}$ for a pair of solutions to some equation $\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{py}\text{'}\mathbf{+}\mathbf{qy}\mathbf{=}\mathbf{0}$ (with $\mathbf{p}$ and $\mathbf{q}$ continuous).

(a) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{6}{\mathbf{e}}^{\mathbf{4}\mathbf{t}}$

(b) $\mathbf{w}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{3}}$

Use Abel's formula (Problem) to determine (up to a constant multiple) the Wronskian of two solutions on $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$ to $\mathbf{ty}\text{'}\text{'}\mathbf{+}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{y}\text{'}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{0}$.

Given that $\mathbf{1}\mathbf{+}\mathbf{t}\mathbf{,}\mathbf{1}\mathbf{+}\mathbf{2}\mathbf{t}$, and $\mathbf{1}\mathbf{+}\mathbf{3}{\mathbf{t}}^{\mathbf{2}}$ are solutions to the differential equation $\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\text{'}\mathbf{+}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{g}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$, find the solution to this equation that satisfies $\mathbf{y}\left(\mathbf{1}\right)\mathbf{=}\mathbf{2}$, $\mathbf{y}\text{'}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{0}$ .

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.

${\mathbf{t}}^{\mathbf{2}}\mathbf{z}\text{'}\text{'}\mathbf{+}\mathbf{tz}\text{'}\mathbf{+}\mathbf{9}\mathbf{z}\mathbf{=}\mathbf{-}\mathbf{tan}\mathbf{\left(}\mathbf{3}\mathbf{lnt}\mathbf{\right)}$

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.

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

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.

${\mathbf{t}}^{\mathbf{2}}\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{tz}\mathbf{\text{'}}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{t}\left(\mathbf{1}\mathbf{+}\frac{\mathbf{3}}{\mathbf{lnt}}\right)$

The Bessel equation of order one-half ${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}\left({\mathbf{t}}^{\mathbf{2}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\right)\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}$ has two linearly independent solutions, ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{cost}\mathbf{,} {\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{sin}$.

Find a general solution to the non-homogeneous equation.

${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}\left({\mathbf{t}}^{\mathbf{2}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\right)\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{5}\mathbf{/}\mathbf{2}}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}$

.

Find a second linearly independent solution using reduction of order.

${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}\mathbf{;} \mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{1}}$.

A differential equation and a nontrivial solution f are given. Find a second linearly independent solution using reduction of order.

${\mathbf{t}}^{\mathbf{2}}\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{6}\mathbf{ty}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}\mathbf{;} \mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{-}\mathbf{2}}$

Find a second linearly independent solution using reduction of order.

$\mathbf{tx}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{x}\mathbf{\text{'}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}\mathbf{;} \mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{t}}$

Find a second linearly independent solution using reduction of order.

$\mathbf{ty}\text{'}\text{'}\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{t}\mathbf{)}\mathbf{y}\text{'}\mathbf{+}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{t}\mathbf{>}\mathbf{0}\mathbf{;} \mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{t}}$

Find a particular solution to the non-homogeneous equation $\mathbf{ty}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{2}\mathbf{t}}$, given that $\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{t}}$ is a solution to the corresponding homogeneous equation.

Find a particular solution to the non-homogeneous equation $(1-\mathrm{t})\mathrm{y}\text{'}\text{'}+\mathrm{ty}\text{'}-\mathrm{y}=(1-\mathrm{t}{)}^2$, given that  $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{t}$is a solution to the corresponding homogeneous equation.

In quantum mechanics, the study of the Schrödinger equation for the case of a harmonic oscillator leads to a consideration of Hermite's equation, $\mathrm{y}\text{'}\text{'}-2\mathrm{ty}\text{'}+\mathrm{\lambda y}=0$ , where $\mathrm{\lambda }$ is a parameter. Use the reduction of order formula to obtain an integral representation of a second linearly independent solution to Hermite's equation for the given value of $\mathrm{\lambda }$ and the corresponding solution $\mathrm{f}\left(\mathrm{t}\right)$.

$$\begin{gathered} \left(\mathrm{a}\right)\mathrm{\lambda }=4, \mathrm{f}\left(\mathrm{t}\right)=1-2{\mathrm{t}}^2 \\ \left(\mathrm{b}\right) \mathrm{\lambda }=6,\mathrm{f}\left(\mathrm{t}\right)=3\mathrm{t}-2{\mathrm{t}}^3 \end{gathered}$$

Complete the proof of Theorem  8 by solving equation (16).

The reduction of order procedure can be used more generally to reduce a homogeneous linear $(\mathbf{n}\mathbf{-}\mathbf{1})$  th-order equation to a homogeneous linear  $(\mathbf{n}\mathbf{-}\mathbf{1})$th-order equation. For the equation, $\mathbf{ty}\text{'}\text{'}\text{'}\mathbf{-}\mathbf{ty}\text{'}\text{'}\mathbf{+}\mathbf{y}\text{'}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}$ which has $\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^t$ as a solution, use the substitution $\mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{v}\left(\mathbf{t}\right)\mathbf{f}\left(\mathbf{t}\right)$ to reduce this third-order equation to a homogeneous linear second-order equation in the variable $\mathbf{w}\mathbf{=}\mathbf{v}\text{'}$.

The equation $\mathbf{ty}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{t}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}$has $\mathbf{f}\left(\mathbf{t}\right)\mathbf{=}\mathbf{t}$ as a solution. Use the substitution $\mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{v}\left(\mathbf{t}\right)\mathbf{f}\left(\mathbf{t}\right)$ to reduce this third-order equation to a homogeneous linear second-order equation in the variable $\mathbf{w}\mathbf{=}\mathbf{v}\mathbf{\text{'}}$.

The reduction of order formula $\left(13\right)$  can also be derived from Abel’s identity (Problem 32). Let $\mathrm{f}\left(\mathrm{t}\right)$ be a nontrivial solution to  and  a second linearly independent solution. Show that ${\left(\frac{\mathrm{y}}{\mathrm{f}}\right)}^{\text{'}}=\frac{\mathrm{W}[\mathrm{f},\mathrm{y}]}{{\mathrm{f}}^2}$and then use Abel's identity for the Wronskian $\mathrm{W}\left[\mathrm{f},\mathrm{y}\right]$  to obtain the reduction of order formula.

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

Using the paradigm (13), what are the inertia, damping, and stiffness for the equation $\mathrm{y}-6{\mathrm{y}}^2=0$? If y > 0, what is the sign of the “stiffness constant”? Does your answer help explain the runaway behaviour of the solutions $\mathrm{y}\left(\mathrm{t}\right)=\frac{1}{{\left(\mathrm{c}-\mathrm{t}\right)}^2}$?

Try to predict the qualitative features of the solution to $\mathrm{y}-6{\mathrm{y}}^2=0$ that satisfies the initial conditions $\mathrm{y}\left(0\right)=-1,\mathrm{y}\text{'}\left(0\right)=-1$. Compare with the computer-generated Figure 4.23. [Hint: Consider the sign of the spring stiffness]

Show that the three solutions $\frac{1}{{\left(1-\mathrm{t}\right)}^2},\frac{1}{{\left(2-\mathrm{t}\right)}^2},$and $\frac{1}{{\left(3-\mathrm{t}\right)}^2}$ to ${\mathrm{y}}^{\text{'}\text{'}}-6{\mathrm{y}}^2=0$are linearly independent on (-1, 1). (See Problem 35, Exercises 4.2, page 164.)

(a) Use the energy integral lemma to derive the family of solutions $\mathrm{y}\left(\mathrm{t}\right)=\frac{1}{\left(\mathrm{t}-\mathrm{c}\right)}$to the equation $\mathrm{y}=2{\mathrm{y}}^3$.

(b) For $\mathrm{c}\ne 0$show that these solutions are pairwise linearly independent for different values of c in an appropriate interval around t = 0.

(c) Show that none of these solutions satisfies the initial conditions $\mathrm{y}\left(0\right)=1,\mathrm{y}\text{'}\left(0\right)=2$.

Use the energy integral lemma to show that motions of the free undamped mass-spring oscillator $\mathbf{my}\mathbf{"}\mathbf{+}\mathbf{ky}\mathbf{=}\mathbf{0}$ obey  $\mathbf{m}{(\mathbf{y}\mathbf{\text{'}})}^2\mathbf{+}{\mathbf{ky}}^2\mathbf{=}\mathbf{constant}$.