Answer to Question #246023 in Differential Equations for meruem

Question

 Find the eigenvalues and eigenfunctions of the following Sturm-Liouville probelm (e^(2x)y')' + e^(2x) (λ + 1)y = 0; y(0) = 0 = y(π). 

Answer 1

$(e^{2x}y')'+e^{2x}(\lambda +1)y=0\\ e^{2x}y''+2e^{2x}y''+e^{2x}(\lambda +1)y=0\\ \text{Divide through by } e^{2x}\\ y''+2y'+(\lambda + 1)=0\\ \text{We can only have a solution when } \lambda >0\\ m^2+2m+(\lambda + 1)=0\\ m=-1\pm\sqrt{-\lambda}\\ y(x)=e^{-1}[C_1\cos(x\sqrt{\lambda})+C_2\sin(x\sqrt{\lambda})]\\ \text{Apply the boundary conditions}\\ y(0)=e^{-1}C_1 \implies C_1=0\\ y(\pi)=e^{-1}[C_2\sin(\pi \sqrt{\lambda}))\\ C_2 \neq 0\\ \implies \sin(\pi \sqrt{\lambda})=0\\ \implies \pi \sqrt{\lambda}=n\pi ~~~n=1,2,3, \cdots\\ \implies \lambda_n =n^2 \text{This is the Eigen values}\\ y_n(x)=e^{-1}[C_n\sin(nx)]\\ \text{This is the Eigen Functions}$