Oddly Formatted Second Order ODE

[checkmatechamp]

Homework Statement

u'' + w²₀*u = cos(wt)

w refers to omega.

Homework Equations

The Attempt at a Solution

I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something.

Second of all, even if I start solving it, trying to use r² + 1*w²₀ = 0, and then getting that r = w₀, where do I go from there? I have a general solution of y = c₁e^w₀t + c₂e^-w₀t

So then I say that the particular solution has to have the form A*cos(wt) + B*sin(wt), and the second derivative is -A*cos(wt) - B*sin(wt)

So then A*cos(wt) - B*sin(wt) + w₀²*A*cos(wt) + B*sin(wt) = cos(wt)

So then A*cos(wt) + w₀²*A*cos(wt) = 1*cos(wt)

So then that gets me that A + w₀²*A = 1, and A = 1/(w₀²)

The sine terms cancel (and there's no sine in the solution anyway)

So then I have y = c₁e^w₀t + c₂e^-w₀t + (cos(wt))/(w₀²).

The closest choice I see is c₁*cos(w₀t) + c₂*sin(w₀t) + (cos(wt))/w₀², and that choice is wrong.

 

[vela]

Homework Statement

u'' + w²₀*u = cos(wt)

w refers to omega.

Homework Equations

The Attempt at a Solution

I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something.

Second of all, even if I start solving it, trying to use r² + 1*w²₀ = 0, and then getting that r = w₀, where do I go from there? I have a general solution of y = c₁e^w₀t + c₂e^-w₀t

$r=\omega_0$ isn't a solution. Plugging it into the characteristic equation would give you $2\omega_0^2 = 0$, which isn't true in general.

So then I say that the particular solution has to have the form A*cos(wt) + B*sin(wt), and the second derivative is -A*cos(wt) - B*sin(wt)

So then A*cos(wt) - B*sin(wt) + w₀²*A*cos(wt) + B*sin(wt) = cos(wt)

You didn't differentiate correctly, you dropped a sign, and you made an algebra errors distributing $\omega_0^2$.

So then A*cos(wt) + w₀²*A*cos(wt) = 1*cos(wt)

So then that gets me that A + w₀²*A = 1, and A = 1/(w₀²)

The sine terms cancel (and there's no sine in the solution anyway)

So then I have y = c₁e^w₀t + c₂e^-w₀t + (cos(wt))/(w₀²).

The closest choice I see is c₁*cos(w₀t) + c₂*sin(w₀t) + (cos(wt))/w₀², and that choice is wrong.