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Original Message (ID '173951') By Bill Simpson:
If, based on the error messages included in your notebook, I rewrite your problem as In[1]:= a=1/20; b= -1/20; σ=35/100; λ=1; ρ=1; r= 15/1000; DSolve[∂\_x ψ[X]*((X*(b - σ*λ*ρ))-a*X^2)+ ∂\_(X, X)ψ[X]*(1/2*σ^2*X^2)==ψ[X]*r, ψ, X] Out[6]= {{ψ -> Function[{X}, C[1]/X^((2*Sqrt[3]*(-7/(4*Sqrt[3]) - Sqrt[97/3]/4))/7) + C[2]/X^((2*Sqrt[3]*(-7/(4*Sqrt[3]) + Sqrt[97/3]/4))/7)]}} then that manually simplifies to ψ[X_]= C[1] X^(1/2+Sqrt[97]/14)+C[2] X^(1/2-Sqrt[97]/14) if I haven't made any scrape-n-paste errors. When I look at your initial condition of ψ[0]==0 then I am worried. I don't see a way to satisfy that without making the solution==0 and that will then not satisfy your second condition.