# Math Help - 2nd linearly independent solution using reduction of order

1. ## 2nd linearly independent solution using reduction of order

How do you find the second linearly independent solution using reduction of order.

Examples
$

t^{2}y'' + 6ty' +6y = 0
t > 0
f(t) = t^{-2}
$

$

ty'' + (1-2t)y' + (t-1)y = 0
t > 0
f(t) = e^t
$

2. Suppose we have $t^{2}y'' + 6ty' +6y = 0, \ t > 0, \ y_{1}(t) = t^{-2}$.

We know that the second solution will have the form $y_{2}(t) = v(t)y_{1}(t)$. Now:

$y_{2}(t) = t^{-2}v$

$y_{2}'(t) = -2t^{-3}v+t^{-2}v'$

$y_{2}''(t) = \cdots$

Plug these into differential equation. You will get an expression only involving derivatives of $v$. Then make a substitution to get a linear first order differential equation.