Solution to partial differential equation

[mrandersdk]

I have the partial differential equation:

$$\frac{\partial \Psi(z,t)}{\partial t} + a * \cos^2(\theta(z,t)) \frac{\partial \Psi(z,t)}{\partial z} - b \frac{N(z)}{\Omega(t)} \cos^4(\theta(z,t))\Psi(z,t) = 0$$

a,b are constants N(z) and $$\Omega(t)$$ are known functions of z and t respectivly, and $$\theta(z,t)$$ is a known function of z and t. I need to find $$\Psi$$, I've searched on the net but couldn't find a solution.

I guess the general form must be

$$\frac{\partial \Psi(z,t)}{\partial t} + f(z,t) \frac{\partial \Psi(z,t)}{\partial z} - g(z,t)\Psi = 0$$

 

[gtoguy97]

where f(z,t) = a * \cos^2(\theta(z,t)) and g(z,t) = b \frac{N(z)}{\Omega(t)} \cos^4(\theta(z,t)). However, I don't know how to solve this equation. Can someone help me? The solution of this equation can be found using the method of characteristics. The characteristic equations are given by: \frac{dz}{dt}=f(z,t) and \frac{d\Psi}{dt}=g(z,t)\Psi. Solving these equations yields \Psi(z,t)=\Psi_0(z_0)\exp\left(-\int_0^t g(z(s),s)ds\right), where z_0 is the initial position along the characteristic curve at time t=0.