1. ## Cauchy-Euler Dif EQ

For a Cauchy Euler Dif EQ, its auxiliary equation for this Dif EQ:

$\displaystyle at^2y'' + bty' + cy = g(t)$ is:

$\displaystyle am^2 + (b-a)m + c = 0$.

Use this auxiliary equation to solve the Cauchy-Euler equation below:

$\displaystyle t^2y'' - 5ty' + 8y = 0$ subject to $\displaystyle y(2) = 32$ and $\displaystyle y'(2) = 0$

2. Originally Posted by alikation0
For a Cauchy Euler Dif EQ, its auxiliary equation for this Dif EQ:

$\displaystyle at^2y'' + bty' + cy = g(t)$ is:

$\displaystyle am^2 + (b-a)m + c = 0$.

Use this auxiliary equation to solve the Cauchy-Euler equation below:

$\displaystyle t^2y'' - 5ty' + 8y = 0$ subject to $\displaystyle y(2) = 32$ and $\displaystyle y'(2) = 0$
Well, a = 1, b = -5 and c = 8, so the auxiliary equation becomes
$\displaystyle m^2 - 6m + 8 = 0$

$\displaystyle (m - 2)(m - 4) = 0$

So m = 2 and m = 4.

Thus the most general solution to this equation is
$\displaystyle y = Ax^2 + Bx^4$

-Dan

3. If you want a solution on $\displaystyle (0,\infty)$ then it is like topsquark said. If you want a solution on the open set $\displaystyle (-\infty,0)\cup (0,\infty)$ then it is $\displaystyle A|x|^2+B|x|^4$.