Show that if y1 is a solution to the ODE y'''+a2y''+a1y'+a0y=0 then the substitution y=uy1 reduces the order of the equation to a 2nd order linear ODE.
The Attempt at a Solution
well, I calculated first, second and third derivatives of y and plugged them in the equation and after cancellation and some tedious algebraic operations I obtained this new equation:
(3y1''+2a2y1'+ a1y1)u' + (3y1' + a2y1)u'' + u'''y1=0
Now I'm stuck and don't know what I should do next at this point. We still have the third derivative of u in the equation and I don't know how to cancel that.