Help solving non homogenous second order ODEs

In summary, the conversation is about using the method of undetermined coefficients to solve a differential equation. The problem given is 2y'' + 3y' + y = t^2, and the speaker has guessed Y = At^2 as a potential solution. However, this does not solve the equation. Another person suggests using a particular solution of the form Y=At^2+Bt+C, which leads to a system of 3 equations with 3 unknowns. The method of undetermined coefficients relies on all linear combinations of the linearly independent derivatives of the RHS, and it is important to consider lower power terms on the LHS that may have canceled out. By plugging in the suggested Y, a system of
  • #1
seang
184
0
I'm using the method of undetermined coefficients here, but I'm either not making the correct ansatz or I'm just confused on the method.

The problem is 2y'' + 3y' + y = t^2.

I gussed Y = At^2. Is this correct? It doesn't solve the differential equation, which is the only check I know.

So from there (assuming Y is correct) I plug in and get 4A +6At + At^2 = t^2. Now I don't know what to do next.
 
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  • #2
Try a particular solution of the form Y=At^2+Bt+C where A,B,C are constants. You'll get 3 equations with 3 unknowns to solve.
 
  • #3
The method of undetermined coefficients relies on all linear combinations of the linearly independent derivatives of the RHS. You really don't know yet whether there are lower power terms on the LHS that have simply canceled out. So your Y should be Y(t) = At^2 + 2Bt + 2C = At^2 + Dt + E. Plug this Y into the original equation and you will get a system of 3 linear equations that is easily solved for the coefficients A, D and E.
 

FAQ: Help solving non homogenous second order ODEs

How do I determine the general solution for a non-homogeneous second order ODE?

In order to determine the general solution for a non-homogeneous second order ODE, you can use the method of undetermined coefficients. This involves finding a particular solution by guessing a function that satisfies the non-homogeneous term, and then adding it to the general solution of the homogeneous equation.

Can I use the method of variation of parameters for non-homogeneous second order ODEs?

Yes, the method of variation of parameters can also be used to solve non-homogeneous second order ODEs. This method involves finding a particular solution by multiplying the general solution of the homogeneous equation by a set of functions, and then using a variation of these functions in the solution.

Are there any special cases for solving non-homogeneous second order ODEs?

Yes, there are two special cases for solving non-homogeneous second order ODEs: when the non-homogeneous term is a constant and when it is a polynomial with all roots being distinct. In these cases, the particular solution can be found by simply plugging in the non-homogeneous term into the general solution of the homogeneous equation.

What is the difference between a non-homogeneous and a homogeneous second order ODE?

A homogeneous second order ODE has a non-zero homogeneous term, which is equal to zero. This means that the equation only contains derivatives and functions of the dependent variable. A non-homogeneous second order ODE, on the other hand, has a non-zero non-homogeneous term, which contains functions of the independent variable.

Is there a specific method for solving non-homogeneous second order ODEs with constant coefficients?

Yes, for non-homogeneous second order ODEs with constant coefficients, you can use the method of undetermined coefficients. This method involves finding a particular solution by guessing a function that satisfies the non-homogeneous term, and then adding it to the general solution of the homogeneous equation. However, this method is not applicable for all non-homogeneous ODEs and other methods may need to be used.

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