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

**Solution:**

Given: Differential equation is x^{2}(dy/dx) = y - xy, y(-1) = -1

Rearrange the terms in terms of dy/dx

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

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

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

Integrate on both sides, we get

∫(1/y)dy = ∫(1 - x)/x^{2}dx

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

-1/x - ln|x| = ln|y| + c --- (a)

Given: When x = -1, y = -1, substitute in eq(a)

-1/-1 - ln|-1| = ln|-1| +c

1 - 0 = 0 + c

c = 1

The solution is ln|y| = -1/x - ln|x| - 1

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

**Summary:**

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

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