I am having the following system of equations
$u_{tt}$ - $4u_{xx} = 0 $ where $ x \in R , t>0 $
$u(x,0) = 2sin(x) , u_{t}(x,0)=x^2sin(x) , x \in R $
I think it is called Cauchy equation for the wave equation or similar and I tried to find methods how to solve it ,but I didn't find.Can someone explain how such a problem is solved and propose a solution?
One of the available methods is the Laplace transform. Sumarizing the steps,
$$ s^2U(x,s)-u(x,0)-s u_t(x,0)-4 U_{xx}(x,t) = 0 $$
or
$$ s^2U(x,s)-2\sin x-s x^2\sin x-4 U_{xx}(x,t) = 0 $$
now solving for $x$
$$ U(x,s) = e^{-\frac{s x}{2}} \left(c_1(s) e^{s x}+c_2(s)\right)+\frac{\left(s \left(s^2+4\right)^2 x^2+2 s (s (s (s+4)+8)-48)+32\right) \sin (x)+16 s \left(s^2+4\right) x \cos (x)}{\left(s^2+4\right)^3} $$
Now assuming $c_1(s) = c_2(s) = 0$
$$ U(x,s) = \frac{\left(s \left(s^2+4\right)^2 x^2+2 s (s (s (s+4)+8)-48)+32\right) \sin (x)+16 s \left(s^2+4\right) x \cos (x)}{\left(s^2+4\right)^3} $$
and now anti-transforming we get
$$ u(x,t) = \sin (x) \left(\left(4 t^2+x^2\right) \cos (2 t)+\sin (2 t)\right)+4 t x \sin (2 t) \cos (x) $$