# Second Order Linear Differential Equation (Non-constant coefficients)

## 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.

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CAF123
Gold Member
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.

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

CAF123
Gold Member
Yeah. We've only covered linear DE's.
Are you at all familiar with separation of variables?

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