Chapter 8: Ordinary Differential Equations

Solve

Solveby the method used in solving,for the following three cases, to obtain the result.

(a) cis not equal to eitheror b;

(b);$\mathit{a}\mathbf{\ne }\mathit{b}\mathbf{,}\mathbf{}\mathit{c}\mathbf{=}\mathit{a}\mathbf{;}$

(c).$\mathbf{a}\mathbf{=}\mathbf{b}\mathbf{=}\mathbf{c}$

Find the general solutions of the following equations and compare computer solutions.

$\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{+}\mathbf{4}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$Hint: Find the four 4th roots of -4 (see Chapter 2, Section 10).

Consider the differential equation $\left(D-a\right)\left(D-b\right)\mathbf{y}\mathbf{=}{\mathbf{P}}_{\mathbf{n}}\left(x\right)$, where ${\mathbf{P}}_{\mathbf{n}}\left(X\right)$is a polynomial of degree $\mathbf{n}$. Show that a particular solution of this equation is given by $\left(6.24\right)$with $\mathbf{c}\mathbf{=}\mathbf{0}$; that is, ${\mathbf{y}}_{\mathbf{p}}$is $\{\begin{array}{ll}a\mathrm{polynomial}{Q}_n\left(x\right)\mathrm{of}\mathrm{degree}n\mathrm{if}a\mathrm{and}b\mathrm{are}\mathrm{both}\mathrm{different}\mathrm{from}\mathrm{zero};& \\ {\mathrm{xQ}}_n\left(x\right)\mathrm{if}a\not\equiv 0,\mathrm{but}b=0& \\ x^2{Q}_n\left(x\right)\mathrm{if}a=b=0& \end{array}$

Find the general solutions of the following equations and compare computer solutions.

${\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{-}\mathbf{16}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$

For the following problems, verify the given solution and then, by method (e) above, find a second solution of the given equation

$x^2{y}^{\text{'}\text{'}}+(x+1)y^{\text{'}}-y=0$

A solution containing 90% by volume of alcohol (in water) runs at 1 gal/min into a 100-gal tank of pure water where it is continually mixed. The mixture is withdrawn at the rate of 1 gal/min. When will it start coming out 50% alcohol?

Solve the following sets of equations by the Laplace transform method

$$\begin{gathered} \dot{\mathbf{y}}\mathbf{,}\dot{\mathbf{z}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}{\mathit{z}}_{\mathbf{0}}\mathbf{=}\mathbf{1} \\ \mathit{z}\mathbf{-}\dot{\mathbf{y}\mathbf{=}\mathbf{t}} \end{gathered}$$

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)].

${(D-2)}^{2}y=16$

Solve the following differential equations by method (a) or (b) above.${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{2}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$

. Hint: The solution is$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathbf{\text{erf}}\mathit{x}\mathbf{+}{\mathit{c}}_{\mathbf{2}}$see Chapter 11, Section 9 for the definition of $\mathbf{\text{erf}}\mathit{x}$.

Problems 2 and 3, use (12.6) to solve (12.1) when $\mathit{f}\left(t\right)$is as given.$\mathit{f}\left(t\right)\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{\omega }\mathit{t}$