# need some help setting up a few problems

• May 15th 2011, 08:53 PM
Jeonsah
need some help setting up a few problems
Here are the problems I am having difficulty with:

1.
Given the characteristic equation of a 4th order differential equation has repeated roots of i and - i,
a) write the general solution of the differential equation
b)write the original differential equation

2.
$\displaystyle (x-2)y'' + (x-1)y' + 4y = 0$
a) write the general solution near x = 0.
b) find the indicial equation and the exponents at the singularity.

Those are the 2 problems I am having trouble with. Its all just on how to start.. Here is what im am doing for each of them:

1.[/B]
A. I dont know how to do....
B.
$\displaystyle (m^2+2m+1)(m^2+2m+1)$
$\displaystyle y^(^4^) +4y''' + 6y'' + 4y' + 1 = 0$

2.
With this you can just substitute
$\displaystyle \sum Cn X^n$
where n = 0 to infinity and its appropriate derivatives. correct?
• May 15th 2011, 11:43 PM
FernandoRevilla
Quote:

Originally Posted by Jeonsah
1.Given the characteristic equation of a 4th order differential equation has repeated roots of i and - i,
a) write the general solution of the differential equation
b)write the original differential equation

If $\displaystyle a\pm bi\;(b\neq 0)$ are roots of the characteristic equation with multiplicity 2, choose

$\displaystyle e^{at}\cos t,\;e^{at}\sin bt,\;te^{at}\cos t,\;te^{at}\sin bt$

to form a basis of the solution space.