Differential Equations: Second Order Equations

[peace-Econ]

Homework Statement

find a particular solutions for the given differentiable equation.

Homework Equations

y''+5y'+6y=4-t^2

The Attempt at a Solution

Because the right-hand side is a polynomial of degree 1, so I want to have a particular solution of the same form. It's like y=a(t^2)+b, but if so, I kinda stuck in the middle of calculation. 6a=-1 (this is fine) and 2a+10at+6b=4 (I want to take out t). i need to use this technique. Please help me find what's wrong here.

Thank you so much!

 

[Whitishcube]

it seems that your original guess at the particular solution is a bit lacking. your choice for y(t) is missing something... can you guess what it is missing?

 

[peace-Econ]

yea, I realized that, but I don't kno what's missing...that's a problem.

 

[Harrisonized]

You should guess y=at²+bt+c, and I'll even tell you why.

Let y=at²+b

If you plug that into y''+5y'+6y, you get:

2a +5(2at) +6(at²+b) = 4-t²

Now you nothing with which to cancel out that 10at in the middle of the left side.

 

[peace-Econ]

I already tried y=at^2+bt+c, but I still didn't have a way t cancel out t...

 

[peace-Econ]

Oh, wait! I don't have to cancel out t...!lol I was too stupid...