# A problem about reduction of the order of a linear ODE

## Homework Statement

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.

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## Homework Statement

Show that if y1 is a solution to the ODE y+a2y''+a1y'+a0y=0

The first term should be y'''.

then the substitution y=u1 reduces the order of the equation to a 2nd order linear ODE.
Presumably you mean y = uy1.

## 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

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.

I didn't work it all out but, assuming you have checked your work, you can just let v = u' and you will have a second order equation in v.

The first term should be y'''.
Yes.

Presumably you mean y = uy1.
Yes.

I didn't work it all out but, assuming you have checked your work, you can just let v = u' and you will have a second order equation in v.

That's a good idea. but do you think the professor wanted us to do that at this step? I mean does he accept this as the solution to his problem?

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That's a good idea. but do you think the professor wanted us to do that at this step? I mean does he accept this as the solution to his problem?

Of course, you need an "= 0" on the right side to make it an equation. But yes, or course, you have reduced the problem of solving a 3rd order equation to one of solving a 2nd order equation.

Of course, you need an "= 0" on the right side to make it an equation. But yes, or course, you have reduced the problem of solving a 3rd order equation to one of solving a 2nd order equation.

Fine. the next question asks me that If we have two linearly independent solutions y1,y2 for this ODE we can find the 3rd solution. Can I use the idea of order reduction from this problem to solve that? If yes, how?
Thanks for helping.

Fine. the next question asks me that If we have two linearly independent solutions y1,y2 for this ODE we can find the 3rd solution. Can I use the idea of order reduction from this problem to solve that? If yes, how?
Thanks for helping.

I haven't ever tried that since it never comes up. My guess is that uy2 would lead to a second solution to your u equation allowing you to reduce it to a first order. So I expect the answer is yes. I guess all you can do is try it.