Question

Using the Frobenius method, solve the following ODE:

x²y"+4xy'+(x²+2)y=0

Answer 1

Let the solution of above differential equation be

$y=a_ox^m+a_1x^{m+1}+a_2x^{m+2}+......................+a_nx^{n+m}$

$\dfrac{dy}{dx}=ma_ox^{m-1}+(m+1)a_1x^m+(m+2)a_2x^{m+1}+.........................$

$\dfrac{d^2y}{dx^2}=m(m-1)a_ox^{m-2}+(m+1)ma_1x^{m-1}+(m+2)(m+1)a_2x^{m}+..................$

$x^2\dfrac{d^2y}{dx^2}+4x\dfrac{dy}{dx}+(x^2+2)y=0$

$x^2[m(m-1)a_ox^{m-2}+(m+1)ma_1x^{m-1}+(m+2)(m+1)a_2x^{m}+..................]+4x[ma_ox^{m-1}+(m+1)a_1x^m+(m+2)a_2x^{m+1}+.........................]+(x^2+2)[a_ox^m+a_1x^{m+1}+a_2x^{m+2}+......................+a_nx^{n+m}]=0$

Equating the coefficient of lowest degree in x (which is x^m) equal to zero

$m(m-1)a_o+4ma_o+2a_o=0$

$m^2+3m+2=0$

$m=-2,-1$

For further powers of x,

$x^{m+1}$:

$m(m+1)a_1+4(m+1)a_1+2a_1=0$

$m^2+m+4m+4+2=0$

$m^2+5m+6=0$

$m=-2,-3$

Since the equation has three roots it cannot be solved by frobenius method

The given differential equation refers to Sturm-Liouville equation