# Find the general solution of the given differential equation x\frac{dy}{dx}-y=x^2\sin x give

Find the general solution of the given differential equation
$x\frac{dy}{dx}-y={x}^{2}\mathrm{sin}x$
give the largest interval over which the genreral solution is defined.
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Bernard Lacey
$x\frac{dy}{dx}-y={x}^{2}\mathrm{sin}x$
$⇒\frac{dy}{dx}-\frac{y}{x}=x\mathrm{sin}x$
This is 1-st order linear in y so integrating Factor $={e}^{\int -\frac{1}{x}dx}$
$={e}^{\mathrm{ln}\left(\frac{1}{x}\right)}$
$=\frac{1}{x}$
multiplying the given equation with I.F. we get
$\frac{1}{x}\frac{dy}{dx}-\frac{y}{{x}^{2}}=\mathrm{sin}x$
$⇒\frac{d}{dx}\left(\frac{y}{x}\right)=\mathrm{sin}x$
integrating both side we get
$\int d\left(\frac{y}{x}\right)=\mathrm{sin}x$
$⇒\frac{y}{x}=-\mathrm{cos}x+C$
$⇒y=-x\mathrm{cos}x+Cx$
Since y is continuous on $\left(-\mathrm{\infty },\mathrm{\infty }\right)$ so longest inter val of existence is $\left(-\mathrm{\infty },\mathrm{\infty }\right)$
Since $x\mathrm{cos}x$ and Cx both does not goes to 0 as $x\to \mathrm{\infty }$
Hence then is no transient form.

Ella Williams
This will be useful to me. Thanks to.