Question

how do we solve the initial-value problem?

−7y''−7y=sin(x); y(0)=2 y'(0)=−4

Answer 1

are you all in systems of first order eq yet?

Answer 2

yea we passed that section@

Answer 3

do you remember how to set up the matrix, because that looks like what you need, only you have a Q(x) matrix for x'=(C)(x)+Q(x)

Answer 4

I'm sorry but I have no idea to what you are referring to?

Answer 5

My bad, what I meant was have you all learned to take that second order eq and turn it into a system?

Answer 6

Where in that system, that second order becomes two first order equations and then you have them expressed in terms of matrix vectors.

Answer 7

I just found my mistake in L's section, go check it out!

Answer 8

ok will do! thanks so much!

Answer 9

so solve the eq for y''\[y''=y+\sin(x)/7\]
let \[ x_1=y; x_1'=y'=x_2; x_2'=y''\]
so then y'' is \[x_2'=x_1+\sin(x)/7\]
now you have two eqs
\[x_1'=x_2\]
\[x_2'=x_1+\sin(x)/7\]

Answer 10

This becomes a matrix X'=k (x) + f(x)
\[\left(\begin{matrix}x_1' \\ x_2'\end{matrix}\right)=\left[\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right]\left(\begin{matrix}x_1 \\ x_2\end{matrix}\right)+\left(\begin{matrix}0 \\ \sin(x)/7\end{matrix}\right)\]

Answer 11

@daomowon
I already solved it without the matrix in the section L differential equations, just in case you're curious

Answer 12

from there you treat the entire eq as if you would a homogeneous system of first order differential eqs. and solve for eigenvalues and vectors. Then take the Wronskian matrix of your eigenvectors and homogeneous solutions then solve the system with sin(x)/7 involved to get your actual equations by doing variation of parameters. and getting an alternate version of that first order system with respect to u'(x). It is way too hard to write the formula in with the subscripts, but it's reasonable when you look at it like that. Out of all the techniques I've learned, that is the only one I see possible. Good thing that you don't have a complete function matrix. Otherwise your last inegrals would be harder. Check your notes on variation of parameters dealing with systems of nonhomogeneous first order linear systems and that will get you through to a final solution.

Answer 13

where can I see it?

Answer 14

http://openstudy.com/study#/updates/4f35a463e4b0fc0c1a0c91e3

Answer 15

cool thanks