Help simple 2nd order linear ODE

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    2nd order Linear Ode
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SUMMARY

The discussion centers on solving the second-order linear ordinary differential equation (ODE) given by v'' + (2/t)v' + b v = 0, where b is a constant. The user expresses difficulty in solving this nonconstant coefficient ODE, contrasting it with their training focused on constant coefficients. The solution provided by Wolfram Alpha is v(r) = (c_1 e^(-sqrt(-k^2) r))/r + (c_2 e^(sqrt(-k^2) r))/(2 sqrt(-k^2) r). The transformation v = y/t simplifies the equation to y'' + b y = 0, leading to solutions that depend on the sign of b, resulting in either exponential or sinusoidal functions.

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y2kevin
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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.
 
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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)
 

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