Chapter 8: Ordinary Differential Equations

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$\mathbf{(}\mathbf{6}\mathbf{.15}\mathbf{)}$].

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

Solve the differential equation $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}{\mathbf{a}}^{\mathbf{2}}\mathbf{y}\mathbf{=}\mathbf{f}\left(t\right)$, here

$\mathbf{f}\left(t\right)\mathbf{=}\{\begin{array}{l}0,t<0,\\ 1,t>0,\end{array}{y}_0=y{\text{'}}_0=0.$

Hint: Use the convolution integral as in the example:

Solve the following equations using method (d) above.

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}-y=x-x^{-1}$

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by $(\mathbf{6}.\mathbf{18}),(\mathbf{6}.\mathbf{23}),$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$\mathbf{(}\mathbf{6}\mathbf{.26}\mathbf{)}$and the discussion after$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}$].

Hint: First solve.

Use L31 to derive L21.

Find the family of orthogonal trajectories of the circles ${\left(x-h\right)}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}{\mathit{h}}^{\mathbf{2}}$. (See the instructions above Problem 2.31.)

(a)Consider a light beam travelling downward into the ocean. As the beam progresses, it is partially absorbed and its intensity decreases. The rate at which the intensity is decreasing with depth at any point is proportional to the intensity at that depth. The proportionality constant $\mathit{\mu }$is called the linear absorption coefficient. Show that if the intensity at the surface is ${\mathit{I}}_{\mathbf{0}}$, the intensity at a distance s below the surface is $\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{\mu }\mathbf{s}}$. The linear absorption coefficient for water is of the order of ${\mathbf{10}}^{\mathbf{.}\mathbf{2}}\mathbf{}\mathit{f}{\mathit{t}}^{\mathbf{.}\mathbf{1}}$(the exact value depending on the wavelength of the light and the impurities in the water). For this value of μ, find the intensity as a fraction of the surface intensity at a depth of 1 ft, ft,ft,mile. When the intensity of a light beam has been reduced to half its surface intensity $\left(I=\frac{1}{2}{I}_0\right)$, the distance the light has penetrated into the absorbing substance is called the half-value thickness of the substance. Find the half-value thickness in terms of $\mathit{\mu }$. Find the half-value thickness for water for the value of $\mathit{\mu }$given above.

(b) Note that the differential equation and its solution in this problem are mathematically the same as those in Example 1, although the physical problem and the terminology are different. In discussing radioactive decay, we call $\mathit{\lambda }$the decay constant, and we define the half-life T of a radioactive substance as the time when $\mathit{N}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{N}}_{\mathbf{0}}$(compare half-value thickness). Find the relation between $\mathit{\lambda }$and T.

Solve the following equations using method (d) above $x^2{y}^{\text{'}\text{'}}-5x{y}^{\text{'}}+9y=2x^3$.

Find the family of curves satisfying the differential equation $\left(x+y\right)\mathit{d}\mathit{y}\mathbf{+}\left(x-y\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$and also find their orthogonal trajectories.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\text{'}}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{5}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}}\mathbf{t}\mathbf{,}\mathbf{\hspace{1em}}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}$