Find the order and degree of Ry" =[1+(y')^2]^3/2

Answer 1

This is 2nd order nonlinear ODE, but we can integrate it by setting y' = u, then y'' = u' and we get

u' = (1 + u²)^3/2 and we can separate variables: ∫du/(1 + u²)^3/2 = ∫dx;

Substitution: u = tanθ; du = sec²θdθ. So, ∫sec²θdθ/sec³θ = x + C; ∫cosθdθ = x + C; sinθ = x + C;

u/(1 + u²)^1/2 = x + C; u²/(1 + u²) = (x + C)²; u^-2 + 1 = (x + C)^-2;

y' = ±((x + C)^-2 - 1)^-1/2 and y = ±∫((x+ C)^-2- 1)^-1/2dx