Question

Find an explicit solution of the given initial - value problem . x ^ 2 * (dy)/(dx) = y - xy , y(- 1) = - 2

Answer 1

The ODE,

$x^2\dfrac{\mathrm dy}{\mathrm dx}=y-xy$

is separable, since we can write

$\dfrac{\mathrm dy}y=\dfrac{1-x}{x^2}\,\mathrm dx$

Integrate both sides to get

$\ln|y|=-\dfrac1x-\ln|x|+C$

Solve for y :

$y=e^{-\frac1x-\ln|x|+C}$

$y=C\dfrac{e^{-\frac1x}}x$

Use the initial condition to solve for C :

$-2=C\dfrac{e^{-\frac1{-1}}}{-1}=-Ce\implies C=\dfrac2e$

So the particular solution to the IVP is

$y(x)=\dfrac{2e^{-\frac1x}}{ex}=\dfrac{2e^{-\frac1x-1}}x$