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trasher [3.6K]
3 years ago
7

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

Mathematics
1 answer:
BARSIC [14]3 years ago
4 0

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 <em>y</em> :

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

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

Use the initial condition to solve for <em>C</em> :

-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

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