Question

This is for differential equations, but i figure maybe someone here could help me

I need help if anyone has the time for a long problem dealing with differential equations and the undetermined coefficient method...here is the problem:

Solve:
y^(4)+y^(2)=t^3+t+te^(2t)+e^(t)*cos(2t)
given
***y(0)=1, y^(1)(0)=0, y^(2)(0)=-1, y^(3)(0)=1

***y^(1) represents the first derivative, y^(2) the second derivative, and y^(3) the third derivative, respectively

I could use all the help I could get! Thanks

Answer 1

In order to solve a diff eqn using Method of Undetermined Coefficients--that is find y(t), you should remember that y(c), the characteristic eqn, and y(p), the particular solution, need to be found.

This is just a brief explanation of what you have to do. I've included links to help you understand the process better.

First, you need to find the characteristic eqn, y(c). In order to do that, apply the Method of Characteristics, that is, set the side with the y's equal to 0 and write the equation as a polynomial--then, find the roots/zeros of that polynomial.

Second, you need to find the particular solution, y(p). To do that, you have to look at the RHS of your original diff. eqn and guess an eqn of the same form/type as the RHS. I'm sure you know there are tables that can help you guess your y(p) based on the function you're basing your guess on. If the y(p) you guess is the same exact one as your y(c), you have to make adjustments to your y(p). For example, let us say my y(c) = C1*e^(2t) and I guess my y(p) = A*e^(2t) then I have to multiply my y(p) by a t so that it's not the same as y(c).

Anyway, after finding y(p), you have to take it's derivatives up to the highest order of derivation in your original diff. eqn. In your case, you have to find all derivatives up to the 4th derivative of y(p). Then, you'll substite y(p), y'(p)...etc into the diff. eqn.
For example:
\[ y _{p}^{(4)}+y _{p}'' = t^3+t+te^{2t}+e^{t}\cos(2t)\] After substitution y(p), you'll be able to solve for the coefficients C1, C2 etc. Finally, after finding your coefficients, you can write your y(p).

Third, you need to combine the two to get y(t) = y(c) + y(p)

Your diff. eqn is a 4th order but is handled in a similar fashion to the 2nd order ones. Here's a really good example of a 4th order example. If you can follow what is being done, you'll be all set. If you still need help, I'll be here to assist you :]

http://tutorial.math.lamar.edu/Classes/DE/HOUndeterminedCoeff.aspx

Another link:
http://www.math.binghamton.edu/grads/kjones/courses/m371summer08/docs/uc.pdf

Answer 2

take laplace transfom for both sides