A problem about reduction of the order of a linear ODE

[AdrianZ]

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.

 

[LCKurtz]

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.

 

[AdrianZ]

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?

 

[LCKurtz]

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.

 

[AdrianZ]

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.

 

[LCKurtz]

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.