Finding general solution of linear ODE inhomogeneous

Click For Summary

Homework Help Overview

The discussion revolves around finding the general solution to a linear inhomogeneous ordinary differential equation (ODE) of the form y'' - 2y' - 24y = 50e^(6x) - 14cos(x) - 175sin(x). Participants are exploring the methods for determining particular solutions and addressing issues related to characteristic equations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the factorization of the characteristic equation and the form of the particular solution. Questions arise regarding the correct coefficients and the handling of complex roots in relation to the undetermined coefficients method.

Discussion Status

There is an ongoing exchange of ideas regarding the correct approach to finding the general solution. Some participants have offered guidance on the form of the particular solution and the implications of the characteristic roots. Multiple interpretations of the problem are being explored, with no explicit consensus reached yet.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the amount of information they can share or the methods they can use. There are indications of confusion regarding the treatment of certain terms in the solution process.

ohspyro89
Messages
13
Reaction score
0

Homework Statement


Find the general solution to y''-2y'-24y=50e6x-14cos(x)-175sin(x)


Homework Equations


I can't figure out how to solve for B,C,D, and E. I'm wondering if I did something wrong.



The Attempt at a Solution



I'm attaching photos, since it'd take forever to type this all out. I'm stuck on this one, and I'm working on another problem at the same time.

IMAG0209.jpg

IMAG0210.jpg

IMAG0211.jpg
 
Physics news on Phys.org
it should be factored as (m-6)(m+4)
not (m-6)(m-4) ill look at the rest.
 
Yes, you did something wrong. Your particular solution should be
yp = Axe6x + Bsin(x) + Ccos(x)

The general solution will be y = c1e6x + c2e-4x + yp
 
*Face-Palm*

Alright, I'll work on it. Thanks guys!

Also, was I right to get rid of one of the +-i for the undetermined coefficients part? So, now I don't have a D and E to worry about, but it covers the cosx-sinx.
 
You really don't need either of the +/- i pairs. If your roots of the char. equation include +/- i, the solutions will include e^(ix) and e^(-ix), but you can take a linear combination of these and work with cos(x) and sin(x) instead.
 
How's this? I got a constant to be zero, which seems odd to me. Is this the right general solution?

IMAG0212.jpg
 
A should equal 5
 
Looks reasonable - you can check it yourself. Is y'' - 2y' - 24y = 50e6x-14cos(x)-175sin(x) an identity? IOW, if you replace the 3 terms on the left with your solution and its first two derivatives, do you end up with what you have on the right?
 

Similar threads

  • · Replies 12 ·
Replies
12
Views
2K
  • · Replies 5 ·
Replies
5
Views
1K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 3 ·
Replies
3
Views
1K
Replies
1
Views
1K
  • · Replies 2 ·
Replies
2
Views
1K