Question

Solve the differential equation de x[dx/dy+y]=1-y​

Answer 1

This is linear in y so we rewrite it in the standard form y' + P(x) y = Q(x):

dy/dx + y = (1 - y) /x yields dy/dx + (1 + 1//x) y = 1/x. so that P = 1 + 1/x and Q = 1/x.

Then the integrating factor is v(x) = e ^ {Integral P(x) dx] = e^ [x + ln x] = x e^x.

The solution is given by y.v = Integral [Q.v}

ie. y (xe^x) = Integral[e^x dx] = e^x + c.

Finally y = [1 + c e^(-x) ] / x.