Find an explicit solution of the given initial-value problem. x²(dy/dx) = y - xy, y(-1) = -1

Solution:

Given: Differential equation is x²(dy/dx) = y - xy, y(-1) = -1

Rearrange the terms in terms of dy/dx

dy/dx = y(1 - x)/x²

(1/y)dy/dx = (1 - x)/x²

(1/y)dy = (1 - x)/ x²dx

Integrate on both sides, we get

∫(1/y)dy = ∫(1 - x)/x²dx

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. x²(dy/dx) = y - xy, y(-1) = -1

Summary:

The explicit solution of the given initial-value problem. x²(dy/dx) = y - xy, y(-1) = -1 is ln|y| = -1/x - ln|x| - 1.