Solve using separation of variables and find particular solution(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} - u = 0 \ for\ 0 <x<1, t >0 [/tex]

[tex] \frac{\partial u}{\partial t} (x,0) = 0 [/tex]

[tex] 0,t) = u(1,t) = 0 [/tex]

to assume u(x,t) = X(x) T(t)

then [tex] \frac{X''}{X} = \frac{T'' - T}{T} = \lambda [/tex]

solving for X(x) yields [tex] X(x) = C_{1} \cos{\sqrt{\lambda} x} + C_{2} \sin{\sqrt{\lambda} x} [/tex]

and i get C1 = 0 and C2 assumed to be 1 and [itex] \lambda = \sqrt{n \pi} [/itex]

now for T(t),

[tex] T'' - (1 + n^2 pi^2) T = 0 [/tex]

now here is where i am stuck...

is the answer just the

[tex] C_{1} \cos{\sqrt{1 + n^2 \pi^2} t}+ C_{2} \sin{\sqrt{1 + n^2 \pi^2} t} [/tex]

i'm not quite sure what to do after this

the answer in my book is [tex] \cos{\sqrt{n^2 \pi^2 -1}} t} \sin (n \pi x) [/tex]

i got the x part right... but what about the t part

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# Partial DIfferential Equations

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