Show that the substitution u = y' leads to a bernoulli equation. solve this equation (see section 2.5). xy'' = y' + (y' )3

If , then and we have



Divide both sides by , so that we get



This is the Bernoulli equation we wanted to find. So we can substitute . This gives , so the ODE is equivalent to



which is linear in . Multiply both sides by and rearrange terms a bit to get



and now the LHS contains a derivative of a product, namely



Now you can solve for , then integrate that result to find .