Cauchy-Euler Dif EQ

alikation0 · #1 $Old$ October 21st, 2007, 10:55 AM

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

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

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

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

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

topsquark · #2 $Old$ October 21st, 2007, 11:02 AM

Quote:

Originally Posted by alikation0 $View Post$

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

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

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

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

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

Well, a = 1, b = -5 and c = 8, so the auxiliary equation becomes
$m^2 - 6m + 8 = 0$

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

So m = 2 and m = 4.

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

-Dan

ThePerfectHacker · #3 $Old$ October 21st, 2007, 12:50 PM

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