Chapter 8: Ordinary Differential Equations

Solve Example 4 using the general solution $y=a\sinh x+b\cosh x$.

Use L34 and L2 to find the inverse transform of $\mathbf{G}\left(p\right)\mathbf{H}\left(p\right)$whenand $\mathbf{G}\left(p\right)\mathbf{=}\mathbf{1}\mathbf{/}\left(p+a\right)\mathbf{}\mathbf{and}\mathbf{}\mathbf{H}\left(p\right)\mathbf{=}\mathbf{1}\mathbf{/}\left(p+b\right)$; your result should be L7 .

By using $L2$, verify $L7$and$L8$ in the Laplace transform table.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.${\mathit{y}}^{\mathbf{t}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{t}}^{\mathbf{t}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{2}\mathit{t}\mathbf{=}\mathbf{3}$

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

2. $\mathbf{x}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}}\mathbf{dx}\mathbf{+}\mathbf{y}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{dy}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{}$when$\mathbf{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$

(a) Show that

$\left(D-a\right){\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\left(c-a\right){\mathbf{e}}^{\mathbf{cx}}\mathbf{;}\left(D^2+5D-3\right){\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\left(c^2+5c-3\right){\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\mathbf{L}\left(D\right){\mathbf{e}}^{\mathbf{cx}}\mathbf{,}$where $\mathbf{L}\left(D\right)$is any polynomial in $\mathbf{D}\mathbf{;}\left(D-c\right){\mathbf{xe}}^{\mathbf{cx}}\mathbf{=}{\mathbf{e}}^{\mathbf{cx}}\mathbf{;}\mathbf{}{\left(D-c\right)}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\mathbf{2}{\mathbf{e}}^{\mathbf{cx}}$.

(b) Define the expression $\mathbf{y}\mathbf{=}\left[1/L\left(D\right)\right]\mathbf{u}\left(x\right)$to mean a solution of the differential equation $\mathbf{L}\left(D\right)\mathbf{y}\mathbf{=}\mathbf{u}$.

Using part (a), show that;

localid="1659340707727" $$\begin{gathered} \frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{a}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{cx}}}{\mathbf{c}\mathbf{-}\mathbf{a}}\mathbf{,}\mathbf{}\mathbf{c}\mathbf{\ne }\mathbf{a}\mathbf{;}\frac{\mathbf{1}}{{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathbf{D}\mathbf{-}\mathbf{3}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{cx}}}{{\mathbf{c}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathbf{c}\mathbf{-}\mathbf{3}}\mathbf{;}\frac{\mathbf{1}}{\mathbf{L}\mathbf{\left(}\mathbf{D}\mathbf{\right)}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{cx}}}{\mathbf{L}\mathbf{\left(}\mathbf{c}\mathbf{\right)}}\mathbf{,}\mathbf{}\mathbf{L}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{\ne }\mathbf{0}\mathbf{;} \\ \frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{a}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}{\mathbf{xe}}^{\mathbf{cx}}\mathbf{;}\frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{a}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{.} \end{gathered}$$

(c) The expressionsin (b) are called inverse operators. They can be used to find particular solutions of differential equations. As an example consider localid="1659340713408" $$\begin{gathered} \mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{D}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}\mathbf{,}} \\ \mathbf{y}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{D}\mathbf{-}\mathbf{2}}{\mathbf{e}}^{\mathbf{2}\mathbf{x}\mathbf{,}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{-}\mathbf{2}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{4}} \end{gathered}$$,

Using inverse operators, find particular solutions of Problems 4 to 20. Be careful to use parts 4 or 5 of (b) ifis a root of the auxiliary equation. For example,$\frac{\mathbf{1}}{\left(D-a\right)\left(D-c\right)}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{a}}\frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{a}}{\mathbf{e}}^{\mathbf{cx}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{D}\mathbf{-}\mathbf{c}}\frac{{\mathbf{e}}^{\mathbf{cx}}}{\mathbf{c}\mathbf{-}\mathbf{a}}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{ex}}}{\mathbf{c}\mathbf{-}\mathbf{a}}\mathbf{.}\mathbf{,}$

For the following problems, verify the given solution and then, by method (e) above, find a second solution of the given equation

$x(x+1)y^{\text{'}\text{'}}-(x-1)y^{\text{'}}+y=0$

Solve the following sets of equations by the Laplace transform method

.$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{z}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0} \\ \mathbf{2}\mathit{y}\mathbf{-}\mathit{z}\mathit{z}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{t}\mathbf{}\mathbf{}\mathbf{}{\mathit{z}}_{\mathbf{0}}\mathbf{=}\mathbf{1} \end{gathered}$$

Find the general solutions of the following equations and compare computer solutions.

${\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{0}$

If P dollars are left in the bank at interest I percent per year compounded continuously, find the amount A at time t. Hint: Find dA, the interest on A dollars for time dt.