A problem about reduction of the order of a linear ODE

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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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LCKurtz
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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.
 
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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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LCKurtz
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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.
 
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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.
 
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LCKurtz
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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.
 

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