Laplace transform of the wave equation I started of with the wave equation $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$
with boundary conditions $u=0$ at $x=0$ and $x=1$ and initial condition $u=sin(\pi x)$ and $\frac{\partial u}{\partial t}=0$ when $t=0$
I have gotten to the stage $$\frac{d^2 v}{dx^2}-s^2v=-s* sin(\pi x)$$
Where do I go from here?
My notes say find the complimentary function using the axillary function $m^2-s^2$
to get $$v_c=Ae^{sx}+Be^{-sx}$$ Then it says find the particular integral for the right hand side and use the trial function $$v_p=a*sin(\pi x)+b*cos(\pi x)$$
But I do not quite understand this. 
Why do we need the complimentary function? 
What is the trial function and why do we need it?

$$\frac{d^2 v}{dx^2}-s^2v=-s* sin(\pi x)$$
Once you have this, you have a differential equation of variable $x$. So you will solve it assuming $s$ is a constant. What you have is a linear second order constant coefficient inhomogeneous ordinary differential equation. It will have a complementary (homogeneous) and a particular component.
The auxiliary (characteristic) equation is actually $1-s^2=0$, so the solution will be a linear combination of the exponentials of the solutions of the characteristic equation. Solutions are $\pm s$, exponentials of the solutions are $e^s$ and $e^{-s}$. $A$ and $B$ are constant coefficients.
For the particular solution, the method you describe is using the method of undetermined coefficients. It is pointless to explain it here, there are quite a few things you need to be careful about and you need to study it properly. You can't solve partial differential equations without knowing such fundamentals about ordinary differential equations. But to find $a$ and $b$, you will insert $v_p$ into the equation (keep in mind $s$ is constant) and you will equate coefficients of $\sin$'s and $\cos$'s
