Chapter 8: Ordinary Differential Equations

Use L32 and L3 to obtain L11

$\mathit{L}\left\{tsinat\right\}\mathbf{=}\frac{\mathbf{2}\mathbf{a}\mathbf{p}}{{\left(p^2+a^2\right)}^{\mathbf{2}}}$

Solve the differential equation $\mathit{y}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{0}$by changing from variables role="math" localid="1655272385100" $y,x$to $r,x$ where ${\mathit{y}}^{\mathbf{2}}\mathbf{=}{\mathit{r}}^{\mathbf{2}}$; then $\mathit{y}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{n}\mathbf{\text{'}}\mathbf{-}\mathit{x}$.

Solve the following equations either by method (d)above or by assuming$y=x^k$(or try both methods to compare them). See Problem 15.

$$\begin{gathered} \left(\mathit{a}\right){\mathit{x}}^2{\mathit{y}}^{\mathit{\text{'}}\mathit{\text{'}}}+3\mathit{x}{\mathit{y}}^{\mathit{\text{'}}}-3\mathit{y}=0 \\ \left(\mathit{b}\right){\mathit{x}}^2{\mathit{y}}^{\mathit{\text{'}}\mathit{\text{'}}}+\mathit{x}{\mathit{y}}^{\mathit{\text{'}}}-4\mathit{y}=0 \\ \left(\mathit{c}\right){\mathit{x}}^2{\mathit{y}}^{\mathit{\text{'}}\mathit{\text{'}}}+7\mathit{x}{\mathit{y}}^{\mathit{\text{'}}}+9\mathit{y}=0 \\ \left(\mathit{d}\right){\mathit{x}}^2{\mathit{y}}^{\mathit{\text{'}}\mathit{\text{'}}}-\mathit{x}{\mathit{y}}^{\mathit{\text{'}}}+6\mathit{y}=0 \end{gathered}$$

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

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{cos}\mathbf{}\mathbf{3}\mathbf{t}\mathbf{,}\mathbf{\hspace{1em}}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{2}\mathbf{,}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$

Consider solving an equation like (10.1) but with nonzero initial conditions.

(a) Write the corrected form of (10.2), writing the transfer function in factored form as indicated just after (10.2). Consider the extra terms inwhich arise from the initial conditions; show that the inverse transforms of such terms can always be found from$\mathit{L}\mathbf{}\mathbf{6}\mathbf{,}\mathit{L}\mathbf{}\mathbf{7}\mathbf{,}\mathit{L}\mathbf{}\mathbf{8}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{L}\mathbf{}\mathbf{18}$

(b) Find the explicit form of the inverse transform of the transfer function for$\mathit{a}\mathbf{\ne }\mathit{b}$use (L,7)and so write the general solution of (10.2) with nonzero initial conditions as a convolution integral plus the terms which you found in (a).

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

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{9}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{30}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{x}$

If an incompressible fluid flows in a corner bounded by walls meeting at the origin at an angle of 60', the streamlines of the flow satisfy the equation $\mathbf{2}\mathit{x}\mathit{y}\mathit{d}\mathit{x}\mathbf{+}\left(x^2-y^2\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$. Find the streamlines.

Use L32 and L11 to obtain$\mathit{L}\left(t^{2}sinat\right)$.

Solve the following equations using method (d) above. $x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}-16y=8x^4$

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{18}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{23}\mathbf{)}\mathbf{,}$ or.$\mathbf{(}\mathbf{6}\mathbf{.24}\mathbf{)}$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$\mathbf{(}\mathbf{6}\mathbf{.26}\mathbf{)}$ andthe discussionafter$\mathbf{(}\mathbf{6}\mathbf{.15}\mathbf{)}$].

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{16}\mathit{y}\mathbf{=}\mathbf{16}\mathit{c}\mathit{o}\mathit{s}\mathbf{4}\mathit{x}$