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## Homework Statement

Show that if y

_{1}is a solution to the ODE y'''+a

_{2}y''+a

_{1}y'+a

_{0}y=0 then the substitution y=uy

_{1}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:

(3y

_{1}''+2a

_{2}y

_{1}'+ a

_{1}y

_{1})u' + (3y

_{1}' + a

_{2}y

_{1})u'' + u'''y

_{1}=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.

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