A Bernoulli differential equation is one of the form dy/dx+P(x)y=Q(x)y^n. y(x)=?

Question

A Bernoulli differential equation is one of the form

dy/dx+P(x)y=Q(x)y^n.

Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y^(1−n) transforms the Bernoulli equation into the linear equation

du/dx+(1−n)P(x)u=(1−n)Q(x).

Use an appropriate substitution to solve the equation

y′−(2/x)y=y^3/x^2,

and find the solution that satisfies y(1)=1.

y(x)=

Answer 1

Substitution v = y^{1 - 3} = y^{- 2}; v' = - 2y^{- 3}y'; y' = - v'y³/2;

-v'y³/2 - (2/x)y = y³/x²; v'/2 + (2/x)v = - 1/x²; v' + (4/x)v = - 2/x²;

Integrated facor M = e^∫4/xdx = e^4lnx = x⁴; x⁴v' + 4x³v = - 2x²; d/dx(vx⁴) = - 2x²; vx⁴ = - 2x³/3 + C;

v = - 2/3x^{- 1} + Cx^{- 4} ; 1/y² = - 2/3x^-1 + Cx^{- 4}. .