How to Solve a System of Coupled Differential Equations?

[Saxonphone]

Homework Statement

2. Homework Equations [/b

Homework Statement

I have to find a(t) and b(t), using the following set of equations:

$$\frac{\hbar\omega_{1}}{2}a(t)\cos{\alpha}+\frac{\hbar\omega_{1}}{2}b(t) exp(-i\omega t) \sin{\alpha}=i\hbar\frac{da}{dt}$$
$$\frac{\hbar\omega_{1}}{2}a(t) exp(-i\omega t) \sin{\alpha}-\frac{\hbar\omega_{1}}{2}b(t)\cos{\alpha}=i\hbar\frac{db}{dt}$$

knowing that:
$$a(t=0)=\cos{\frac{\alpha}{2}}$$
$$b(t=0)=\sin{\frac{\alpha}{2}}$$

The Attempt at a Solution

Well, I'm stuck. I found a solution to b(t) by eliminating a(t) through multiplication by constants and subtracting thw equations, but not the solution i`m supposed to find (i know the answer, but it is to long, so at least for now i`m not going to write it).
I need some light, how should I approach this?

 

[mighty2000]

Hello! It seems like you are trying to solve a system of coupled differential equations. In this case, it is helpful to use a technique called the method of undetermined coefficients. This involves assuming a solution of the form a(t) = A exp(iωt) and b(t) = B exp(iωt), where A and B are constants that you can solve for.

By substituting these assumed solutions into the given equations, you can solve for A and B. This will give you the general solution for a(t) and b(t). Then, you can use the initial conditions (a(t=0) and b(t=0)) to find the specific values of A and B.

I hope this helps! Let me know if you have any further questions. Good luck with your problem!