Chapter 8: Ordinary Differential Equations

By integrating the appropriate formula with respect to, verify L19.

$L\left(\frac{\sin at}{t}\right)={\tan }^{-1}\left(\frac{a}{p}\right)$

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\left(3.9\right)$, and Example 1.

$\mathit{y}\mathbf{\text{'}}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathbf{+}\mathit{y}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{x}$

Use the convolution integral to find the inverse transforms of:

$\frac{\mathbf{p}}{\left(p+a\right){\left(p+b\right)}^{\mathbf{2}}}$

$\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathit{x}\mathit{y}\mathbf{=}\mathit{y}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{1}$when $\mathit{x}\mathbf{=}\mathbf{1}$.

Find the position x of a particle at time t if its acceleration is$\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{=}\mathit{A}\mathbf{}\mathit{t}\mathbf{}\mathit{\omega }\mathbf{}\mathit{s}\mathit{i}\mathit{n}$.

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)].

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{e}}^{\mathbf{x}}$

Question: The differential equation of a hanging chain supported at its ends is

${\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{2}}\mathbf{=}{\mathbf{k}}^{\mathbf{2}}\left(1+y^{\text{'}2}\right)$. Solve the equation to find the shape of the chain.

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\left(3.9\right)$, and Example .

$\mathit{y}\mathbf{+}\mathit{y}\mathbf{/}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{/}\left(x+\sqrt{x^2+1}\right)$

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)].

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{6}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{12}{\mathit{e}}^{\mathbf{-}\mathbf{x}}$

By replacing$a$in L2 by$a+ib$and then by$a-ib$, and adding and subtracting the results and verify L13 and L14.

$\left(e^{-at}\sin bt\right)$