General solution od PDE of second order

Question

I have following PDE equation: $\frac{\partial^2T(s,x)}{\partial x^2}=\frac{s}{a}T(x,s)$. Why general solution is found by form: $T=c_1 e^{\sqrt{\frac{s}{a}}x}+c_2 e^{-\sqrt{\frac{s}{a}}x}$. I know that it was from $\lambda^2=\frac{s}{a}$ but it is not clear for me where is last equation from( I don't understand proof and evaluation of this) ?

Answer 1

Thank you All! It is derived from Euler's assumption about form of partial solutions: $$T = e^{\lambda(s) x}.$$ Second partial derivative of $T$ in this case will be:$$\frac{\partial T(x,s)}{\partial x^2 } = \lambda^2 e^{\lambda x} = \lambda^2 T$$ Last expression equals $\lambda^2 T = \frac{s}{a}T$, from which we have that $$\lambda_{1,2} = \sqrt{\frac{s}{a}}$$ Genereal solution is the sum of product of partial solutions and some coefficients: $$T = C_1 T_1+C_2 T_2 = C_1 e^{\sqrt{\frac{s}{a}} x}+C_2 e^{-\sqrt{\frac{s}{a}} x}$$