Solve the following differential equation.

Question

Solve the following PDE $$\frac{\partial^{2}z}{\partial x^{2}}+z=0,$$ given that when $x=0$, $z=e^{y}$ and $\frac{\partial z}{\partial x}=1$.

Answer 1

In essential the equation is an ode: the general solution of $z_{xx}+z=0$ is $z=C_1\cos x+C_2\sin x$ where the coefficients are functions of $y$. Substituting the initial condition yields $C_1=e^y$ and $C_2=1$. So the solution is $z(x,y)=e^y\cos x+\sin x$.