Second Order Linear Differential Equation (Non-constant coefficients)

[SherlockOhms]

Homework Statement

Find the general solution for the following differential equation:
y'' + x(y')^2 = 0.

Homework Equations

Integration, differentiation...

The Attempt at a Solution

Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify such an equation to a first order DE. The (y')^2 part is throwing me off as this would give the equation:
v(x)' + x(v(x))^2 = 0.
This would be grand if it weren't for the v(x)^2 term. Any ideas on how to get around this? Thanks.

 

[CAF123]

Are you worried about the v^2 term because it makes the eqn non-linear? If so, there are ways to solve such an equation.

 

[SherlockOhms]

Yeah. We've only covered linear DE's.

 

[CAF123]

Yeah. We've only covered linear DE's.

Are you at all familiar with separation of variables?

 

[SherlockOhms]

Yeah. So I'll have v'(x) = (-x)(v(x))^2 and I can then separate the variables and solve. Thanks!