Find an explicit solution of the given initial-value problem. x² dy/dx = y - xy, y(-1) = -5

Solution:

x² dy/dx = y - xy [Given]

It can be written as

x² dy/dx = y(1 - x)

dy/dx = y (1 - x)/ x² 

dy/y = (1 - x)/ x² . 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. x² dy/dx = y - xy, y(-1) = -5

Summary:

The explicit solution of the given initial-value problem x² dy/dx = y - xy, y(-1) = -5 is y(x) = 25 e^{-1/x - 1/5} /x.