91Ó°ÊÓ

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)].

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

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 .

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

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

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

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

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

$\mathit{y}\mathbf{"}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{4}$, $\underset{\mathbf{0}}{\mathbf{∑}}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{ψ}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathbf{2}$

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

Use the convolution integral to find the inverse transforms of:$\frac{\mathbf{p}}{\left(p+a\right)\left(p^2+b^2\right)}$

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

Verify L15 to L18, by combining appropriate preceding formulas

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 $\mathit{v}\mathbf{→}\mathit{c}$as $\mathit{t}\mathbf{→}\mathbf{∞}$. Find the distance travelled by the electron in timeif it starts from rest.