I have the following system of initial value problem:
$$\begin{bmatrix} y_1 \\y_2 \end{bmatrix}''-\lambda^2\begin{bmatrix} y_1 \\y_2 \end{bmatrix}=\begin{bmatrix}0 \\0 \end{bmatrix};\qquad y_2(0)=y_1(1)=0$$ where $\lambda=i\beta$ for some $\beta>0$.
If I treat them as separate differential equations $y_1''-\lambda^2y_1=0$ and $y_2''-\lambda^2y_2=0$, I get the solution $$\begin{bmatrix} y_1 \\y_2 \end{bmatrix}=\begin{bmatrix} c_1\cos(\beta x)+c_2\sin(\beta x) \\c_3\cos(\beta x)+c_4\sin(\beta x) \end{bmatrix}$$ with $4$ arbitrary constants.
My question: Is there an alternate way to solve the system so that I only get $2$ arbitrary constants?
Motivation of question: Since I am only given $2$ boundary conditions, I was hoping to only have $2$ constants to solve for.