# Math Help - Nonhomogeneous wave equation

1. ## Nonhomogeneous wave equation

Hi, how should I start to solve the following PDE?

$u_{tt} - u_{xx} = sin(x) cos(5x) sin(w t)$ 0<x<pi, t>0

u(0,t)=u(pi, t)=0
u(x,0)= $u_{t}(x,0)$=0.

My aim is actually to find the values for w, for which the solution does not diverge. Is there a way to find it without completeley solving the problem.
Thanks for the help.

2. Originally Posted by Gaudium
Hi, how should I start to solve the following PDE?

$u_{tt} - u_{xx} = sin(x) cos(5x) sin(w t)$ 0<x<pi, t>0

u(0,t)=u(pi, t)=0
u(x,0)= $u_{t}(x,0)$=0.

My aim is actually to find the values for w, for which the solution does not diverge. Is there a way to find it without completeley solving the problem.
Thanks for the help.
Yes. If you look for solutions in the form

$
u = \sum_{n=1}^{\infty} T_n(t) \sin n \pi x
$

and sub into you PDE noting that $\sin x \cos 5x = \frac{1}{2} \sin 6x - \frac{1}{2} \sin 4x$, then comparing terms of $\sin n \pi x$ you get

$
T_n'' + n^2 \pi^2 T_n = 0\;\; \text{for}\; n \ne 4 \; \text{and}\;6
$
and

$
T_4'' + 16 \pi^2 T_4 = - \frac{1}{2} \sin \omega t
$

and

$
T_6'' + 36 \pi^2 T_6 = \frac{1}{2} \sin \omega t
$

Now the solution of the first is bounded and then you ask, for what values of $\omega$ are the solutions of the second and third unbounded?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8781975507736206, "perplexity": 705.8370109393619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737937342.66/warc/CC-MAIN-20151001221857-00085-ip-10-137-6-227.ec2.internal.warc.gz"}