Answer to Question #2783 in Differential Equations for Ahmad

Question

solve my differential equation
y'' =(y')[sup]3[/sup] + x[sup]2[/sup]

Answer 1

Let’s solve the homogeneous equation:
y''- y'³ = 0
Denote y' as p:
p'p - p³ = 0
dp/dy = p²
p = -1/x + C
Let’s solve nonhomogeneous equitation:
p'p - p³ = 1/p²
p'p³ - p⁵ = 1
dp/dx = 1/(p (C - p)²) + p²
dx = (p(C - p)²)/(1 + p³(C - p)²) dp
The last equitation hasn’t analytical solution.