Chapter 8: Ordinary Differential Equations

Question: The curvature of a curvein the $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$plane is

$\mathbf{K}\mathbf{=}{\mathbf{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}{\left(1+y^{\text{'}2}\right)}^{\mathbf{-}\mathbf{3}\mathbf{/}\mathbf{2}}$

With K = const., solve this differential equation to show that curves of constant curvature are circles (or straight lines).

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{\mathbf{1}}{\mathbf{3}}\mathit{c}\mathit{m}$has radius $\mathbf{0}\mathbf{.}\mathbf{4}\mathit{c}\mathit{m}$after 6 months, how long will it take:

(a) For the radius to be $\frac{\mathbf{1}}{4}\mathit{c}\mathit{m}$?

(b) For the volume of the mothball to be half of what it was originally?

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

${\mathit{y}}^{\mathbf{"}}\mathbf{-}\mathbf{6}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathit{t}{\mathit{e}}^{\mathbf{3}\mathbf{t}}$,

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{2}\mathbf{x}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{-}\mathbf{y}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{0}$when $\mathit{x}\mathbf{=}\sqrt{\mathbf{2}}$.

The momentum pof an electron at speednear the speedof light increases according to the formula $\mathbf{p}\mathbf{=}\frac{\mathbf{mv}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$, whereis a constant (mass of the electron). If an electron is subject to a constant force F, Newton’s second law describing its motion is localid="1659249453669" $\frac{\mathbf{d}\mathbf{p}}{\mathbf{d}\mathbf{x}}\mathbf{=}\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\frac{\mathbf{m}\mathbf{v}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}\mathbf{=}\mathit{F}\mathbf{.}$

Find $\mathit{v}\left(t\right)$and show that $\mathit{v}\mathbf{\to }\mathit{c}$as $\mathit{t}\mathbf{\to }\mathbf{\infty }$. Find the distance travelled by the electron in timeif it starts from rest.

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

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{D}\mathbf{+}\mathbf{6}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$

$\mathit{y}\mathit{d}\mathit{y}\mathbf{+}\left(x{y}^2-8x\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{3}$ when $\mathit{x}\mathbf{=}\mathbf{1}$.

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 $\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{9}\mathbf{)}$, and Example .

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

Verify L15 to L18, by combining appropriate preceding formulas

Question: Solve${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}{\mathit{\omega }}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{0}$by method (c) above and compare with the solution as a linear equation with constant coefficients.