Higher Order Linear Homogenous Differential Equation

[Mark Rice]

Homework Statement

Hi, basically I have a boundary value problem and just want to check that my general solution is correct.

x'''' + 16x = 0

Homework Equations

The Attempt at a Solution

I'm pretty sure you make a characteristic equation which would be m⁴ + 16 = 0.
Solving this I get m to be √2 +- √2 i and -√2 +- √2 i. I therefore get my general solution to be:

Ae^(√2t)cos(√2t) + Be^(-√2t)cos(√2t) + Ce^(-√2t)sin(√2t) + De^(√2t)sin(√2t)

Is this correct or am I on totally the wrong track? I just want to make sure this is correct before applying the initial and boundary coniditions. Thanks.

 

[SteamKing]

Homework Statement

Hi, basically I have a boundary value problem and just want to check that my general solution is correct.

x'''' + 16x = 0

Homework Equations

The Attempt at a Solution

I'm pretty sure you make a characteristic equation which would be m⁴ + 16 = 0.
Solving this I get m to be √2 +- √2 i and -√2 +- √2 i. I therefore get my general solution to be:

Ae^(√2t)cos(√2t) + Be^(-√2t)cos(√2t) + Ce^(-√2t)sin(√2t) + De^(√2t)sin(√2t)

Is this correct or am I on totally the wrong track? I just want to make sure this is correct before applying the initial and boundary coniditions. Thanks.

Take your general solution and plug it back into the original ODE. If you're on the right track, you'll know it when you get zero on both sides of the equation.

 

[Mark44]

Take your general solution and plug it back into the original ODE. If you're on the right track, you'll know it when you get zero on both sides of the equation.

What SteamKing suggests is something you should always do when you're working with diff. equations.

 

[Mark Rice]

Right cool thanks guys!

 

[Mark Rice]

Cool, I plugged it in and got 0 (99% sure I did this correctly, there was a lot of terms haha!), so that means it is the correct and I can move on to applying the boundary functions? Thanks for all the help, will definitely use that plugging in method to double check my answers in future :)

 

[SteamKing]

Yep, plug ahead.