Find an explicit solution of the given initial-value problem. x²(dy/dx) = y - xy, y(-1) = -4

Solution:

Given: Differential equation is x²(dy/dx) = y - xy, y(-1) = -4

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 = -4, substitute in eq(a)

-1/-1 - ln|-1| = ln|-4| +c

1 - 0 = 1.38 + c

c = -0.38

The solution is ln|y| = -1/x - ln|x| + 0.38

Find an explicit solution of the given initial-value problem. x²(dy/dx) = y - xy, y(-1) = -4

Summary:

The explicit solution of the given initial-value problem. x²(dy/dx) = y - xy, y(-1) = -4 is ln|y| = -1/x - ln|x| + 0.38