
wave equation
The question is Solve the one dimensional wave equation
$\displaystyle \frac{1}{c^2}$ http://www.mathhelpforum.com/mathhe...363cc0f41.gif
on $\displaystyle 0<=x<=3$, subject to the conditions
$\displaystyle u(x,0) = \frac{1}{2}x$, $\displaystyle u_t(x,0)=x(3x), u(0,t)=u(3,t)=0$
well i started it, but im not sure if this is right coz i don't really understand wave or heat equations SIGH...
$\displaystyle
u_tt(x,t) = X''(x)T(t) $ and $\displaystyle u_tt(x,t) = X(x)T''(t)
$
$\displaystyle \frac{1}{c^2} $ http://www.mathhelpforum.com/mathhe...363cc0f41.gif
Case 1 let $\displaystyle v = w^2 > 0 $
$\displaystyle
\frac{X''(x)}{X(x)} = w^2 $
$\displaystyle
X''(x) = w^2 X(x) = 0 $
Using Auxilliary Equations:
$\displaystyle
m^2  w^2 = 0$
So $\displaystyle m = + w $
And $\displaystyle X(x) = C_1 e^{wx} + C_2 e^{wx}$
As $\displaystyle u(0,t) = u(3,t) = 0, X(0) = V(3) = 0$
Therefore, $\displaystyle C_1 + C_2 = 0$ and $\displaystyle C_1 =  C_2$
Therefore, no solutions...
Case 2 $\displaystyle v = w^2=0$
$\displaystyle X''(x)  0X(x) = 0$
Auxilliary Equations $\displaystyle m^2  0 = 0$, $\displaystyle m = + 0 $
Therefore, $\displaystyle X(x) = C_3 e^0 + C_4 e^{0} $
As $\displaystyle X(0) = X(2) = 0, X(0) = C_3, X(3) = 2C_3 + C_4 $
So $\displaystyle 2C_3 + C_4 = 0 $
Therefore, no solution as $\displaystyle 2C_3 = C_4$
Case 3 $\displaystyle v = w^2$ as $\displaystyle v<0$
$\displaystyle
\frac{X''(x)}{X(x)} = w^2 $
$\displaystyle X''(x) + w^2X(x) = 0 $
Auxilliary Equation: $\displaystyle m^2 + w^2 = 0, m = +iw $\
So $\displaystyle C_5coswx +C_6sinwx = 0$
$\displaystyle C_5 = 0, $ So $\displaystyle
C_5cos3w + C_6sin3w = 0$
$\displaystyle sin3w = 0$
Therefore, $\displaystyle 3w = \frac{n\Pi}{3}$ and $\displaystyle V = \frac{n^2\Pi^2}{9}$
Thus, $\displaystyle X(x) = B sin $ $\displaystyle \frac{xn\Pi}{3}$
Now Im lost, am i doing it right so far, if so, then how do i go about completing this, coz i have no idea what to do now.. Thank you

Case 1: correct.
Case 2: you have come to the right result but via the wrong method. For case 2 you have:
$\displaystyle \frac{d^2 X}{dx^2} = 0$
so integrating twice gives
$\displaystyle X = A x +B$
and to satisfy BCs $\displaystyle A = B = 0$ hence no solution as you concluded (for the wrong reasons).
Case 3: you have correctly found that:
$\displaystyle X_n (x) = B_n \sin \left( \frac{n \pi}{3} x \right)$
Consequently
$\displaystyle \frac{d^2 T}{dt^2} + c^2 \left(\frac{n \pi}{3}\right)^2 T = 0$
so then, after finding the auxiliary equation, we find the solution for $\displaystyle T$ is
$\displaystyle T_n = C \sin \left(\omega_n c t \right) + D \cos \left(\omega_n c t \right)$
where $\displaystyle \omega_n = \frac{n \pi}{3}$ so the general solution is given by:
$\displaystyle u(x,t) = \sum_{n=0}^{n=\infty} a_n(\sin \left(\omega_n c t \right) + b_n \cos \left(\omega_n c t \right)) \sin \left( \omega_n x \right)$
Applying the initial conditions on u(x,0) and u'(x,0) we have
$\displaystyle \sum_{n=1}^{n=\infty} c_n \sin \left( \omega_n x \right) = \frac{1}{2} x$
where $\displaystyle c_n = a_n b_n$ and
$\displaystyle \sum_{n=1}^{n=\infty} a_n \omega_n c \sin \left( \omega_n x \right) = x(3x)$.
In the first of these summations you can find $\displaystyle c_k$ by multiplying by $\displaystyle \sin( \omega_k x)$ on both sides and then integrating wrt x from 0 to 3. Doing this should give you that:
$\displaystyle c_n = (1)^{n+1} \frac{3}{n \pi}$ .
I may have made arithmetic errors here so double check the result in case.
For the second summation it's exactly the same procedure to give you $\displaystyle a_n$. At the end substitute back into your general solution the results for $\displaystyle a_n$ and $\displaystyle c_n = a_n b_n$ and you're done!
An alternative, and easier way would have been to transform the problem to Fourier space and then solve.
Hope that helped!

that's awesome!! Thank you SO much :)