How Do You Solve a Second Order Linear Differential Equation?

[captainjack2000]

Homework Statement

I have the equation -(ck^2)U(k,t)=d^2/dt U(k,t). And I need to find the general solution.

Homework Equations

The Attempt at a Solution

I can rearrange this into the form d^2 U/dt^2 + (ck^2)U(k,t)=0 but I am not sure of the form of the solution to this equation.

could someone please give me pointer in the right direction

 

[captainjack2000]

Is it just U(k,t) = Acos(ckt)+Bcos(ckt)?

 

[aurora14421]

Yeah, it's just the differential equation for SHM so that is the solution.

 

[dx]

No it's not. You will get a constant when you do the integration with respect to t, and this constant can depend on k. So the general solution is what you wrote + f(k), where f(k) is an arbitrary function of k.

EDIT: You should have (√c)k instead of ck in the sin and cos.

 

[dx]

Oops! I'm extremely sorry, my previous post is wrong. The constants A and B are the constants of integration, and they should be arbitrary functions of k.