# Reduction to single higher-order linear ODE

• June 30th 2012, 06:53 PM
fractal5
Reduction to single higher-order linear ODE
I need to reduce the following system to a single higher-order linear ODE for y

x'(t) - y'(t) + z(t) = 0
x(t) + y'(t) - z'(t) = 1
y(t) + z'(t) = t3- 1

I've tried setting ( z = y' - x' ) & (x = 1 - y' - z')
However on substituting these it ends up as a big mess and I can't go anywhere. Could you someone tell me what substitution or technique I should use to work this problem?

Thanks.
• July 1st 2012, 03:39 AM
BobP
Re: Reduction to single higher-order linear ODE
Can get you to a cubic, don't know if it's possible to do better

Differentiate eq(1), $x'' - y''+z'=0$ and substitute from eq(3), $x''-y''+(t^{3}-1-y)=0..........(4)$

Substitute for $z'$ in eq(2) also, $x+y'-(t^{3}-1-y)=1...........(5).$

Now differentiate $(5)$ twice and then subtract $(4)$ to eliminate the $x''$ term.

Note that you lose information during this process though, The 1 on the RHS of the second equation differentiates out and so could be any number at all.
• July 1st 2012, 09:55 AM
fractal5
Re: Reduction to single higher-order linear ODE
Yes that works, thanks a lot. I can't see a way of keeping the "1" unless I integrate the first equation and sub x(t) in the second. But I don't think that's possible. I believe this answer is sufficient. Thanks again :)