Second order linear differential equation

[dvvv]

Homework Statement

Solve the following second order linear differential equation
d²x/dt² + x = 2 cos(t)
subject to the initial condition x(0) = 0 and dx/dt (0) = 0. What type of motion do you find?

Homework Equations

The Attempt at a Solution

I don't know where to start

 

[HallsofIvy]

If you honestly have no idea how to even start a homework problem, you have a serious problem. I recommend you talk to your teacher about this. Obviously, he/she expects you to know how to do problems like this. If you don't your teacher needs to suggest some review.

The standard method of solving a "non-homogeneous linear differential equation with constant coefficients" is first to solve the associated homogeneous equation:
$$\frac{d^2x}{dt^2}+ x= 0$$
Can you do that?

And then look for a single solution to the entire equation. Normally, with "cos(x)" on the left, I would recommend trying something of the form "Acos(x)+ Bsin(x)" but for this equation, as you should be able to see after solving the associated homogeneous equation, that will not work. Instead try something of the form y= Ax cos(x)+ Bx sin(x).

 

[dvvv]

I think I would substitute x=e^rt. I did that on another question.
I'm probably wrong. It's too late for me to learn all this now. Thanks anyway!

 

[jjlg]

Is a movement with external forced f(t)=cos(t), without damping, (can be a spring with external force)