# Homework Help: Nonhomogeneous Equations; Method of Undetermined Coefficient

1. Feb 14, 2010

### Jamin2112

I'm stuck on just one problem.

1. The problem statement, all variables and given/known data

2y'' + 3y' + y = t2 + 3sin(t)

2. Relevant equations

It says in the lesson that if you have a polynomial, guess a solution is

"Yi(t)= Ts(A0tn + A1tn-1 + .......... + An)

where s is the smallest nonnegative integer (s=0,1, or 2) that will ensure that no terms in Yi(t) is a solution of the corresponding homogeneous equation."

And I don't really understand that jargon. Maybe someone could dumb it down for me.

3. The attempt at a solution

Since the solution is the sum of two different functions, t2 and 3sin(t), I can solve the differential equation for each individually, and them add them together at the end since the sum of the solutions to a differential equation is also a solution.

I got the solution to
2y'' + 3y' + y = 3sin(t).

It is Y1(t) = (-3/10)sin(t) -(9/10)cos(t)

But Y2(t) is giving me trouble.

I guessed Y2(t) = A0t2 + A1t + A2 but ended up with a system equations that still had t in it; thus I couldn't solve it.

Set me on the right track.

2. Feb 14, 2010

### rock.freak667

y=A+Bt+Ct2+Dsint+Ecost

then sub that into the different equation and equate coefficients.

3. Feb 14, 2010

### Jamin2112

Does

ring a bell?

4. Feb 14, 2010

### rock.freak667

Sorry, I was used to doing the two together and not separately. Well just substitute y=A0+A1t+A2t2 into the equation. the coefficients of t and the constant on the right side would just be zero.

Though when doing the undetermined coefficient method, I find it best to solve for the homogeneous solution first.

5. Feb 14, 2010

### HallsofIvy

Then show us exactly what you did. Putting that into the left side will give a quadratic and you can set coefficients of corresponding terms equal. (If there is no "corresponding" term on one side, its coefficient is 0.)