Nonlinear differential equation problem.

[center o bass]

Homework Statement

The following equation turned up while I was trying to make an integral
stationary in a 'calculus of variations' problem.

$$y^{\prime}(x)^2 + 1 = y^{\prime\prime}(x) y(x)$$

How would one go about solving this nonlinear equation?

 

[phyzguy]

Equations like this, which do not contain the independent variable (x), can be solved by applying the identity:
$$y'' = \frac{d^2y}{dx^2} = \frac{dy'}{dx} = \frac{dy'}{dy}\frac{dy}{dx} = y' \frac{dy'}{dy}$$
Then you can write:
$$y'^2+1 = y' \frac{dy'}{dy} y$$
Then you collect terms in y and y' to get to a form you can integrate:
$$\frac{y' dy'}{1+y'^2} = \frac{dy}{y}$$
Then integrate both sides and solve for y' in terms of y, and then integrate a second time.