PEEWSRIGWETRYqx

2021-12-21

Find the general solution of the given differential equation.

$x}^{2}{y}^{\prime}+x(x+7)y={e}^{x$

$y=$

enlacamig

Beginner2021-12-22Added 30 answers

The differential equation is an example.

$x}^{2}{y}^{\prime}+x(x+7)y={e}^{x$

On rewriting it, we get

${y}^{\prime}+(1+\frac{2}{x})y=\frac{{e}^{x}}{{x}^{2}}...$(1)

Standard form of the linear differential equation $\frac{dy}{dx}+P(x)y=f(x)$

Clearly equation (1) is in the standard form

Compare the differential equation (1) with standard orm, and identity

$P\left(x\right)=\frac{2+2}{x}$ and $f\left(x\right)=\frac{{e}^{x}}{{x}^{2}}$

The dunctions $P\left(x\right)=\frac{2+2}{x}$ and $f\left(x\right)=\frac{{e}^{x}}{{x}^{2}}$ are continuous on $(0,\mathrm{\infty})$

The integrating facrote is

$e}^{\int P\left(x\right)dx}={e}^{\int (1+\frac{2}{x})dx$

$={e}^{x+2\mathrm{ln}x}$

$e}^{x}{e}^{{\mathrm{ln}x}^{2}$

$={x}^{2}{e}^{x}$

The standard form multiplied by the integrating factor

$x}^{2}{e}^{x}({y}^{\prime}+(1+\frac{2}{x})y)={x}^{2}{e}^{x}\cdot \frac{{e}^{x}}{{x}^{2}$

$x}^{2}{e}^{x}{y}^{\prime}+{x}^{2}{e}^{x}y+2x{e}^{x}y={e}^{2x$

$x}^{2}{e}^{x}\frac{dy}{dx}+{x}^{2}y\frac{d}{dx}\left({e}^{x}\right)+{e}^{x}y\frac{d}{dx}\left({x}^{2}\right)={e}^{2x$

$\frac{d}{dx}\left({x}^{2}{e}^{x}y\right)={e}^{2x}$

${x}^{2}{e}^{x}y=\int {e}^{2x}dx$

${x}^{2}{e}^{x}y=\frac{1}{2}{e}^{2x}+c$

${x}^{2}{e}^{x}y=\frac{1}{2{x}^{2}}{e}^{x}+c{x}^{-2}{e}^{-x}$

Hence the general solution of the given differential equation is

ol3i4c5s4hr

Beginner2021-12-23Added 48 answers

answer
Part 1 $\frac{1}{2{x}^{2}}{e}^{x}+C{x}^{-2}{e}^{-x}$

Part 2$(0,\mathrm{\infty})$

Part 3$C{x}^{-2}{e}^{-x}$

Part 2

Part 3

RizerMix

Expert2021-12-29Added 656 answers

Thank you!!!!!!!

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