Nonhomogeneous Equations; Method of Undetermined Coefficient

[Jamin2112]

I'm stuck on just one problem.

Homework Statement

2y'' + 3y' + y = t² + 3sin(t)

Homework Equations

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

"Y_i(t)= T^s(A₀tⁿ + A₁t^n-1 + ... + A_n)

where s is the smallest nonnegative integer (s=0,1, or 2) that will ensure that no terms in Y_i(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.

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.

 

[rock.freak667]

Just take your PI as

y=A+Bt+Ct²+Dsint+Ecost

then sub that into the different equation and equate coefficients.

 

[Jamin2112]

Just take your PI as

y=A+Bt+Ct²+Dsint+Ecost

then sub that into the different equation and equate coefficients.

I already tried that, bro.

Does

I guessed Y2(t) = A0t2 + A1t + A2

ring a bell?

 

[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.

 

[HallsofIvy]

I'm stuck on just one problem.

Homework Statement

2y'' + 3y' + y = t² + 3sin(t)

Homework Equations

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

"Y_i(t)= T^s(A₀tⁿ + A₁t^n-1 + ... + A_n)

where s is the smallest nonnegative integer (s=0,1, or 2) that will ensure that no terms in Y_i(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.

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.

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.)

Set me on the right track.