Chapter 8: Ordinary Differential Equations

Following the method of equations (10.8)to (10.12), show that ${\mathbf{f}}_{\mathbf{1}}{\mathbf{f}}_{\mathbf{2}}$and ${\mathbf{g}}_{\mathbf{1}}\mathbf{*}{\mathbf{g}}_{\mathbf{2}}$are a pair of Fourier transforms.

Prove the general formula L29.

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.

1.$\mathbf{xy}\mathbf{\text{'}}\mathbf{=}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{3}$, when$\mathbf{x}\mathbf{=}\mathbf{2}$

For integral $k$, verify $L5$and $L6$in the Laplace transform table. Hint: From $L2$you can write: ${\int }_0^{\infty }{e}^{-pt}{e}^{-at}dt=1/(p+a)$. Differentiate this equation repeatedly with respect to $p$. (See Chapter 4, Section 12, Example 4, page 235.) Also note$L32$for the$\Gamma$function results in$L5$and$L6$, see Chapter 11, Problem 5.7.

Continuing the method used in derivingand, verify the Laplace transforms of higher-order derivatives ofgiven in the table (L35).

Verify the statement of Example 2. Also verify that $\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{}\mathit{x}$and$\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{}\mathit{x}$ are solutions of ${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{=}\mathit{y}$.

Solve $\left(12.3\right)$if $\mathit{G}\mathbf{=}\mathbf{0}$and $\mathit{d}\mathit{G}\mathbf{/}\mathit{d}\mathit{t}\mathbf{=}\mathbf{0}$at $\mathit{t}\mathbf{=}\mathbf{0}$ to obtain (12.5). Hint: Use L28 and L3 to find the inverse transform.

Find the general solution of the following differential equations (complementary function particular solution). Find the solution by inspection or by (6.18), (6.23), or (6.24). Also find a computer solution and reconcile differences if necessary, noticing especially whether the solution is in simplest form [see (6.26) and the discussion after (6.15)].

$y^{\text{'}\text{'}}-4y=10$

$y\text{'}\text{'}+yy\text{'}=0$Find a solution satisfying each of the fo♦llowing sets of initial conditions. If your computer says there is no such solution, don’t believe it—do it by hand.

(a)$y\left(0\right)=5,y\text{'}\left(0\right)=0$

(b)$y\left(0\right)=2,y\text{'}\left(0\right)=-2$

(c)$y\left(0\right)=1,y\text{'}\left(0\right)=-1$

(d)$y\left(0\right)=0,y\text{'}\left(0\right)=2$

Find the inverse Laplace transform of ${\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{P}}\mathit{l}\mathbf{}{\mathit{P}}^{\mathbf{2}}$in the following ways:

(a) Using L5 and L27 and the convolution integral of Section 10;

(b) Using L28.