PDE4

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u_{{xx}}-u_{t}=0\,

For this problem, guess that the solution is in the form e^{{ax+bt}}\,. Plug the guess into the original PDE to find that:

a^{2}-b=0\,

a=\pm {\sqrt  {b}}\,

Let b=c_{1}, a constant. Adding together the two cases for a, the homogeneous solution is

u(x,t)=e^{{{\sqrt  {c_{1}}}x+c_{1}t}}+e^{{-{\sqrt  {c_{1}}}x+c_{1}t}}\,

Partial Differential Equations

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