Question

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

I don't have time to do the entire problem, but it looks like this would be pretty easy with laplace transforms

Answer 2

This is an easy problem, but it takes a long time to do.
You can do it with Laplace Transform or with the method of undtermined coefficient. The Laplace Transform can be shorter. If you do not have the answer to test your work. Here it is
\[
y(t) = \frac{t^5}{20}-\frac{5 t^3}{6}+\frac{1}{20}
e^{2 t} t+\frac{23 t}{4}-\\ \frac{9 e^{2
t}}{100}+\frac{1}{10} e^t \sin
(t)-\frac{138 \sin (t)}{25}-\\\frac{1}{5}
e^t \cos (t)+\frac{26 \cos
(t)}{25}+\frac{1}{4}
\]
As you can see from the answer, that it might need time to work it out.