Solving v''+(2/t)v'+bv=0 with nonconstant coefficients

[y2kevin]

Hi, could anyone give me a hint on what method to use to solve this ODE:

v''+(2/t)v'+(b)v=0,

b is a constant and v=v(t).

Most of my ODE training resolves around how to solve the above equation with constant coefficients. AND all of my reference books say that "the nonconstant case is difficult and you won't run into it." Well, here I am, stuck.

The answer wolfram alpha gave is:

v(r) = (c_1 e^(-sqrt(-k^2) r))/r+(c_2 e^(sqrt(-k^2) r))/(2 sqrt(-k^2) r)

No idea on how it got it though.

 

[JJacquelin]

Let v = y/t
which turns the ODE to :
y'' + b y = 0
So, depending the sign of b, the solutions y(t) are sums of exponential or sinusoidal functions of (sqrt(-b)*t) or of (sqrt(b)*t)