\begin{align} \frac{\partial}{\partial x}\psi &=-h\psi+ie^{-it}\phi &&& (1) \\ \frac{\partial}{\partial x}\phi&=h\phi+ie^{it}\psi &&& (2) \\ \frac{\partial}{\partial t}\psi&=\left(-h-\frac{i}{2}\right)\psi-he^{-it}\phi &&& (3) \\ \frac{\partial}{\partial t}\phi&=\left(h+\frac{i}{2}\right)\phi-he^{it}\psi &&& (4) \end{align}
Solution of eqs $(1)$-$(4)$ is,
$$\psi=i\left(C_1e^A − C_2e^{−A}\right)e^{− \frac{i}{ 2}t}$$ $$\phi=\left(-C_1e^{-A} + C_2e^{A}\right)e^{\frac{i}{ 2}t}$$
where $$C_1 = \frac{\left(h − \sqrt{h^2 − 1}\right)^{1/2}} {\sqrt{h^2 − 1}}, \qquad C_2 = \frac{\left(h + \sqrt{h^2 − 1}\right)^{1/2}} {\sqrt{h^2 − 1}}$$ and $$A = \sqrt{h^2 − 1}\big(x + iht\big)$$
How to find these solutions? Anyone can help?