How Do You Solve a Differential Equation Using an Ansatz?

[Hamiltonian]

Homework Statement

Take the $ans\ddot atz$ $x(t) = At cos(ω_{0}t) + Btsin(ω_{0}t)$ and adjust the constants $A, B$
to solve the DE bellow in the case $ω = ω_0$.
$$\ddot x + {\omega_0}^2 x=cos(\omega t)$$

Relevant Equations

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Finding the first and second derivative of out ansatz, $$\dot x(t)=A(cos(\omega_0 t) - t\omega_0 sin(\omega_0 t)) + B(sin(\omega_0 t) + t\omega_0 cos(\omega_0 t))$$ $$\ddot x= A(-2\omega_0 sin(\omega_0 t) - t{\omega_0}^2cos(\omega_0 t)) + B(2\omega_0 cos(\omega_0 cos(\omega_0 t) -t{\omega_0}^2sin(\omega_0 t)))$$
The differential Equation we are trying to find a solution to is, $$\ddot x + {\omega_0}^2 x = cos(\omega_0 t)$$
if we plug in $\dot x$ and $\ddot x$ and after a little simplification we end up with, $$2\omega_0(Bcos(\omega_0 t) - Asin(\omega_0 t)) = cos(\omega_0t)$$
From here we essentially guess A and B such that the LHS=RHS, I can't think of any possible values that could satisfy the equation.

 

[Hill]

I didn't check every step in the algebra, but I see A and B that satisfy the last equation.

 

[Hamiltonian]

I didn't check every step in the algebra, but I see A and B that satisfy the last equation.

I initially thought of $$A=sin(\omega_0 t)$$ and $$B=cos(\omega_0 t)$$ using the trig identity, $cos(2\theta) = cos^2(\theta)-sin^2(\theta)$ we get,
$$2\omega_0 cos(2\omega_0 t) = cos(\omega_0 t)$$
but even this doesn't work.

 

[Hill]

I initially thought of $$A=sin(\omega_0 t)$$ and $$B=cos(\omega_0 t)$$ using the trig identity, $cos(2\theta) = cos^2(\theta)-sin^2(\theta)$ we get,
$$2\omega_0 cos(2\omega_0 t) = cos(\omega_0 t)$$
but even this doesn't work.

A and B are constants!

 

[Mark44]

$2\omega_0(Bcos(\omega_0 t) - Asin(\omega_0 t)) = cos(\omega_0t)$

The above has to be true for all values of the variable t, so it should be clear that the absence of a $\sin(\omega_0 t)$ term on the right side has an effect on that term on the left side.

 

[Hamiltonian]

The above has to be true for all values of the variable t, so it should be clear that the absence of a $\sin(\omega_0 t)$ term on the right side has an effect on that term on the left side.

Are you implying, $A=0$ and $B=\frac{1}{2\omega_0}$

Edit: I shall not show my face around these parts of town henceforth.