Higher order differential equation

[xago]

Homework Statement

Solve the following initial value problem:

2007y⁽⁴⁾-18y⁽³⁾+178y⁽¹⁾ = 0

with initial conditions y(0)=y⁽¹⁾(0)=y⁽²⁾(0)=y⁽³⁾(0)

Homework Equations

Differential equations..

The Attempt at a Solution

From the equation I get r(2007r³ - 18r² +178) = 0

Well first I can't seem to find the roots of the equation (except for r=0) with any method I've been taught and also the fact that they don't actually give any values for the initial conditions but just states that they're equal to each other confuses me. I figure they are just asking for the general solution but I just need to find the roots of the equation which I can't seem to do.

 

[lurflurf]

Do you need an explicit exact solutuion?
r(2007r3 - 18r2 +178) = 0
is messy to solve

Since you have
y(0)=y(1)(0)=y(2)(0)=y(3)(0)
You will have one arbitrary constant in your solution.

 

[xago]

They are not asking to find an exact solution but to just find y(t). I suppose if I could just find out if there's someway to tell if the roots are complex,repeating or just real and distinct I could write out the forumlas for each.

 

[lurflurf]

The roots are approximately
{0,.225977+.386191j,.225977-.386191j,-.442985}
where j^2=-1
The zero root is obvious.
(2007r3 - 18r2 +178)
is optimal when the derivative is zero
6021r^2-36r=0
r={0,4/669}
so
(2007r3 - 18r2 +178)
increases when r<0
decreases when 0<r<4/669
increases when r>4/669
r=0 (2007r3 - 18r2 +178)->178
r=4/669 (2007r3 - 18r2 +178)->~177.998
So we know there is a real root with r>0
and complex conjugate roots