solve second order ODE equation

I try to symbolic the function psi_m(y) and psi_m(x) and solve the equation b with matlab obe solver. The results should come out as equation c.

However my matlab script indicate the error. Could you help me take a look my matlab code below to debug?

Thank you very much!!

clc
clear

syms psi_m(y) psi_m(x) lamda_m  psi(x,y) s_sqar
psi(x,y)=psi_m(y)*sin(lamda_m*x)-psi_m(x)*sin(lamda_m*y)
delta_psi(x,y)=diff(psi(x,y),x,2)+diff(psi(x,y),y,2)

ode=delta_psi(x,y)-s_sqar*psi(x,y)==0

psi_mSol(x)=dsolve(ode)

[Walter Roberson]

I do not think you can have the same function with two different variables

[Torsten]

psi_m (x) = sinh(k_m*x)

is not a solution of the differential equation

psi_m''(x) + (lambda_m^2+s^2)*psi_m(x) = 0, 

but of the differential equation

psi_m''(x) - (lambda_m^2+s^2)*psi_m(x) = 0

if k_m is set to

k_m = sqrt(lambda_m^2+s^2)