Question

Help solve this initial value problem

Answer 1

See did you solve the differential equation first?

Answer 2

@Ozuukai ?

Answer 3

I refuse to do this by hand. Here is the solution for whomever will try it by hand
\[y(x)=\frac{1}{100} \left(-8 \sin ^2(5 x)+20 x \sin (5
x)+15 \sin (5 x)+2 \sin (5 x) \sin (10 x)+\\4 \cos ^3(5
x)-8 \cos ^2(5 x)-296 \cos (5 x)-15 \sin (4 x) \cos
(5 x)\\+10 \sin (6 x) \cos (5 x)+15 \sin (5 x) \cos (4
x)-10 \sin (5 x) \cos (6 x)\right)
\]

Answer 4

The above solution was done using Mathematica

Answer 5

Hello @Ozuukai will you cooperate? Then I will try to help you., :)

Answer 6

\[y''+25y=0\]admits two characteristic solutions of \(y_C=C_1e^{r_1x}+C_2e^{r_2x}\), where the \(r_i\) are roots to the characteristic equation
\[r^2+25=0\implies r_i=\pm 5i\]This means you can generate two independent trigonometric solutions, namely
\[e^{\pm 5ix}=\cos5x\pm i\sin5x\implies Ce^{r_ix}=C_1\cos5x+C_2\sin5x\]That's the easy part.

To solve the nonhomogeneous part, either variation of parameters or the method of undetermined coefficients will serve nicely here. The latter I've always been more familiar with, so that's what I would resort to.

Suppose \(y_p\) is the particular solution to the nonhomogeneous ODE, for which a possible candidate might be \(y_p=a_1+a_2\cos x+a_3\sin x+a_4x\cos5x+a_5x\sin5x\). Then
\[{y_p}''=-a_2\cos x-a_3\sin x-10a_4\sin5x-25a_4x\cos5x\\
\quad\quad\quad\quad+10a_5\cos5x-25a_5x\sin5x\]Substituting into the ODE, you have
\[\begin{align*}
25a_1+24a_2\cos x+24a_3\sin x&\\
\quad\quad\quad-10a_4\sin5x+10a_5\cos5x&=-2+6\sin x+2\cos5x
\end{align*}\]which yields the system
\[\begin{cases}
25a_1=-2\\[1ex]
24a_2=0\\[1ex]
24a_3=6\\[1ex]
-10a_4=0\\[1ex]
10a_5=2
\end{cases}\]etc.

@eliesaab Mathematica must have some weird algorithm for generating the solution it does. If you hit it with `Simplify[]` the resulting solution might get cleaner.

Answer 7

@HolsterEmission you are courageous to do it by hand. I applaud you. If you graph the two soultions, you might find them the same.

Answer 8

Simplified solution by Mathematica
\[
\frac{1}{100} (25 \sin (x)+20 x \sin (5 x)+15 \sin (5
x)-292 \cos (5 x)-8)
\]

Answer 9

@jango_IN_DTOWN apologies i was asleep