Method of undetermined coefficients, when to raise the guess

[Uku]

Hello!

I have some examples of non-homogeneous ODEs to be solved by the undetermined coefficients method. Two from "Pauls math notes" page:

$y''+8y'+16y=e^{-4t}+(t^2+5)e^{-4t}$
The compsol. for this is:
$Y_{c}=C_{1}e^{-4t}+C_{2}te^{-4t}$
The first guess for a particular solution would be:
$Y_{p}=(At^2+Bt+C)e^{-4t}$
When multiplying the brackets with the exponent we get a solution which is in the complementary solution, thus we raise the order, indeed we have to do this twice. Pauls notes (link here) stress that only the part which is the mixed solution needs to be raised. In this example I would then raise only $Bt$ by $t^2$ and $C$ by $t^2$. But the correct form is $t^2(At^2+Bt+C)e^{-4t}$. Wolfram alpha verifies.

While on the other hand, of his other example:
$y''+3y'-28y=7t+e^{-7t}-1$
Complementary solution being $Y_{c}=C_{1}e^{4t}+C_{2}e^{-7t}$
The first guess at a particular solution:
$Y_{p}=At+B+Ce^{-7t}$
Now here the solutions match as well, meaning a raise of the particular matching exponential function:
$Y_{p}=At+B+Cte^{-7t}$ which is the correct guess.

Now... why in the first example is the whole polynomial multiplied by the $t^2$?

And another example:
$y^{(4)}-2y^{(3)}+y''=e^t+1$
The compsol. here is:
$Y_{c}=C_{1}+C_{2}+C_{3}e^{t}+C_{4}te^{t}$
Now, my first guess for a particular solution would be $Ae^t+B$
This is also a solution in the homogeneous equation, so I need to raise it, twice. But according to two sources the correct way of doing this is $At^{2}e^{t}+Bt^2$
Why not raise the function that is actually lapping, meaning $At^{2}e^{t}+B$

So.. that is the question.
Thank you,
U.

 

[algebrat]

There is a form that the driving function (inhomogeneous part) should be into apply thise methods. In some of your examples, the driving function is not of this form, so you break the problem up to find the two particular solutions corresponding to the two driving functions. However, for the third example, you have a fourth derivative so you'll have to be even more careful.

 

[Mandlebra]

Note the repeated roots of the characteristic equation

 

[haruspex]

I can make Paul's procedure consistent if it works like this:
- if a part of the PI matches a part of the CS then multiply it up by t
- if that makes it match another part of the CS, multiply that up too
- apply the above rules iteratively until it settles down

 

[blue_raver22]

Hello U,

The method of undetermined coefficients is a popular technique used to solve non-homogeneous ordinary differential equations (ODEs). It is based on the principle that a particular solution can be found by taking a guess that is similar in form to the non-homogeneous term in the ODE.

In your first example, the non-homogeneous term is a combination of exponential and polynomial functions. When multiplying the polynomial by the exponential, we get a solution that is already in the complementary solution. This means that the guess needs to be raised in order to create a solution that is not already present in the complementary solution. In this case, raising the order of the polynomial by t^2 ensures that the resulting solution is unique and not already present in the complementary solution.

In the second example, the non-homogeneous term is a combination of a linear and exponential function. When taking a guess, we only need to raise the matching exponential function in order to create a unique solution. This is because the linear function does not overlap with the complementary solution.

In your last example, the non-homogeneous term is a combination of exponential and polynomial functions. The correct guess is At^2e^t + B because the polynomial part is not already present in the complementary solution. In this case, raising the order of the exponential function would not create a unique solution.

Overall, the key to determining when to raise the guess in the method of undetermined coefficients is to ensure that the resulting solution is not already present in the complementary solution. I hope this helps clarify the concept for you.