91Ó°ÊÓ

Use the convolution integral to find the inverse transforms of:

$\frac{\mathbf{1}}{\left(p+a\right){\left(p+b\right)}^{\mathbf{2}}}$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$\mathit{y}\mathbf{"}\mathbf{+}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{}\mathbf{}\mathbf{}\mathbf{,}$

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.

$\mathbf{2}\mathit{x}\mathit{y}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{5}\mathbf{/}\mathbf{2}}$

Show that${\mathbf{∫}}_{\mathbf{-}\mathbf{∞}}^{\mathbf{∞}}{\mathit{f}}_{\mathbf{n}}\left(t\right)\mathit{d}\mathit{t}\mathbf{=}\mathbf{1}$for the functions${\mathit{f}}_{\mathbf{n}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$in Figures 11.3 and 11.4.

Solve the following differential equations by method (a) or (b) above.

$x{y}^{\text{'}\text{'}}=y^{\text{'}}+y^{\text{'}3}$

$\left(1+y^2\right)\mathit{d}\mathit{x}\mathbf{+}\mathit{x}\mathit{y}\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{0}$when $\mathit{x}\mathbf{=}\mathbf{5}$.

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)].

$\mathbf{(}\mathit{D}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{−}\mathbf{3}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{24}{\mathit{e}}^{\mathbf{−}\mathbf{3}\mathbf{x}}$

Find the distance which an object moves in time $t$if it starts from rest and has acceleration$\frac{d^{2}x}{d{t}^2}=g{e}^{−kt}$. Show that for small$t$the result is approximately, and for very large, the speed$\frac{dx}{dt}$is approximately $(1.10)$constant. The constant is called the terminal speed . (This problem corresponds roughly to the motion of a parachutist.)

Solve the following differential equations by the methods discussed above and compare computer solutions.

$y^{\text{'}\text{'}}-2y^{\text{'}}=0$

Solve the following differential equations by the methods discussed above and compare computer solutions.

$4y^{\text{'}\text{'}}+12y^{\text{'}}+9=0$