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# Find an explicit solution of the given initial-value problem. x^{2} dy/dx = y - xy, y(-1) = -5

**Solution:**

x^{2} dy/dx = y - xy [Given]

It can be written as

x^{2} dy/dx = y(1 - x)

dy/dx = y (1 - x)/ x^{2}

dy/y = (1 - x)/ x^{2} . dx

First integrate both sides

ln |y| = - 1/x - ln |x| + C

Let us solve for y

y = e^{-1/x - ln|x| + C}

So we get

y = C e^{-1/x }/x

To solve for C let us use the condition

-5 = C e^{-1/-5 }/-5

25 = C e^{1/5 }

C = 25/e^{1/5 }

So the solution is

y(x) = 25/e^{1/5 }. e^{-1/x }/x

y(x) = 25 e^{-1/x - 1/5 }/x

Therefore, the explicit solution is y(x) = 25 e^{-1/x - 1/5 }/x.

## Find an explicit solution of the given initial-value problem. x^{2} dy/dx = y - xy, y(-1) = -5

**Summary:**

The explicit solution of the given initial-value problem x^{2} dy/dx = y - xy, y(-1) = -5 is y(x) = 25 e^{-1/x - 1/5 }/x.

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