 PEEWSRIGWETRYqx

2021-12-21

Find the general solution of the given differential equation.
${x}^{2}{y}^{\prime }+x\left(x+7\right)y={e}^{x}$
$y=$ enlacamig

The differential equation is an example.
${x}^{2}{y}^{\prime }+x\left(x+7\right)y={e}^{x}$
On rewriting it, we get
${y}^{\prime }+\left(1+\frac{2}{x}\right)y=\frac{{e}^{x}}{{x}^{2}}...$(1)
Standard form of the linear differential equation $\frac{dy}{dx}+P\left(x\right)y=f\left(x\right)$
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 $\left(0,\mathrm{\infty }\right)$
The integrating facrote is

$={e}^{x+2\mathrm{ln}x}$
${e}^{x}{e}^{{\mathrm{ln}x}^{2}}$
$={x}^{2}{e}^{x}$
The standard form multiplied by the integrating factor
${x}^{2}{e}^{x}\left({y}^{\prime }+\left(1+\frac{2}{x}\right)y\right)={x}^{2}{e}^{x}\cdot \frac{{e}^{x}}{{x}^{2}}$
${x}^{2}{e}^{x}{y}^{\prime }+{x}^{2}{e}^{x}y+2x{e}^{x}y={e}^{2x}$

${x}^{2}{e}^{x}y=\frac{1}{2}{e}^{2x}+c$
${x}^{2}{e}^{x}y=\frac{1}{2{x}^{2}}{e}^{x}+c{x}^{-2}{e}^{-x}$
Hence the general solution of the given differential equation is ol3i4c5s4hr

answer Part 1 $\frac{1}{2{x}^{2}}{e}^{x}+C{x}^{-2}{e}^{-x}$
Part 2 $\left(0,\mathrm{\infty }\right)$
Part 3 $C{x}^{-2}{e}^{-x}$ RizerMix