First order differential equations

Question

Find an explicit solution to the differential equation by making an appropriate substitution.

x(dy/dx)-(1+x)y=xy^2

Answer 1

x(^dy/_dx)-(1+x)y=xy²

^dy/_dx - (1+x)y/x = y²

This is Bernoulli's First Order ODE. The equation can be put in this form:

^dy/_dx - P(x)y = f(x)yⁿ

Let u = y^1-n = y^1-2= y^-1, Therefore du = (-1/y²)dy

Multiply both sides by -1/y²

(-1/y²)(^dy/_dx) + (1+x)/(xy) = -1

Make the equation in terms of u and x.

du/dx +u(1+x)/x = -1

The integrating factor is e ^∫P(x)dx

e^∫P(x)dx = e ^{∫(1+x)/x dx} =e ^ln(x)+x = xe^x

Therefore:

(xe^x)du/dx +u(e^x)(1+x) = -xe^x

d(uxe^x)/dx = -xe^x (Product rule of differentiation)

uxe^x = - ∫xe^xdx (Use integration by parts)

uxe^x = - xe^x+e^x + C

u = (- xe^x+e^x + C)/xe^x

Substitute u in terms of y:

y^-1 = (- xe^x+e^x + C)/xe^x

y = xe^x/(- xe^x+e^x + C)