Consider the linear second order differential equation:
(x-β)y′′-xy′+γy=0,where β and γ are parameters.
(a) Classify the point x=β with a short explanation.
(b) What is the leading order term in each of the two series solutions about x=β? Discuss the analyticity of the solutions. Do not find the full series solutions. (Hint: you may want to consider a simpletransformation z=f(x) to look at the be haviour around z=0.)
(c) If β=-1 and γ ≠ 0,1,2,3..., determine a series solution, giving a general form for the coefficients in terms of the parameters. Is this solution analytic? Without finding the coefficients, state the form of theother solution. Is this solution ana lytic at x=β?
(d) If γ=m, where m=0,1,2,3..., (still with β=-1) discuss the form of the analytic series solution.
Write out this series solution, corresponding to γ=2, satisfying y(0)=1.
(x-β)y′′-xy′+γy=0,where β and γ are parameters.
(a) Classify the point x=β with a short explanation.
(b) What is the leading order term in each of the two series solutions about x=β? Discuss the analyticity of the solutions. Do not find the full series solutions. (Hint: you may want to consider a simpletransformation z=f(x) to look at the be haviour around z=0.)
(c) If β=-1 and γ ≠ 0,1,2,3..., determine a series solution, giving a general form for the coefficients in terms of the parameters. Is this solution analytic? Without finding the coefficients, state the form of theother solution. Is this solution ana lytic at x=β?
(d) If γ=m, where m=0,1,2,3..., (still with β=-1) discuss the form of the analytic series solution.
Write out this series solution, corresponding to γ=2, satisfying y(0)=1.