91Ó°ÊÓ

Solve

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

(a) cis not equal to eitheror b;

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

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

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

$3x{y}^{\text{'}\text{'}}-2(3x-1)y^{\text{'}}+(3x-2)y=0$

Solve the following sets of equations by the Laplace transform method

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

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 The mixture is withdrawn at the rate of 1 gal/min.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?

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

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

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

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

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 $\{\begin{array}{ll}a\mathrm{polynomial}{Q}_n\left(x\right)\mathrm{of}\mathrm{degree}n\mathrm{if}a\mathrm{and}b\mathrm{are}\mathrm{both}\mathrm{different}\mathrm{from}\mathrm{zero};& \\ {\mathrm{xQ}}_n\left(x\right)\mathrm{if}a≢0,\mathrm{but}b=0& \\ x^2{Q}_n\left(x\right)\mathrm{if}a=b=0& \end{array}$

By using $L2$, verify $L7$and$L8$ in the Laplace transform table.

Solve Example 4 using the general solution $y=a\sinh x+b\cosh x$.

Use L34 and L2 to find the inverse transform of $\mathbf{G}\left(p\right)\mathbf{H}\left(p\right)$whenand $\mathbf{G}\left(p\right)\mathbf{=}\mathbf{1}\mathbf{/}\left(p+a\right)\mathbf{}\mathbf{and}\mathbf{}\mathbf{H}\left(p\right)\mathbf{=}\mathbf{1}\mathbf{/}\left(p+b\right)$; your result should be L7 .