How Do You Solve a Damped Harmonic Oscillator Differential Equation?

[razorfever]

damped harmonic oscillator, urgent help needed!

Homework Statement

for distinct roots (k1, k2) of the equation k^2 + 2Bk + w^2 show that x(t) = Ae^(k1t) + Be^(k2t) is a solution of the following differential equation: (d^2)x/dt^2 + 2B(dx/dt) + (w^2)x = 0

Homework Equations

The Attempt at a Solution

I have no idea where to begin, can anyone point me in the right direction or giv me some sort of outline to follow

 

[Redbelly98]

I would first find k1 and k2. This involves solving the quadratic equation you were given.

 

[nlightner]

?

I understand your urgency and need for help. The damped harmonic oscillator is a system that experiences a damping force and oscillates with a decreasing amplitude over time. It is widely studied in physics and engineering, and has many applications in real-world systems.

To prove that x(t) = Ae^(k1t) + Be^(k2t) is a solution to the given differential equation, we can start by substituting x(t) into the equation and simplifying. We know that x(t) is a function of time, so we can rewrite it as x(t) = x(t), and then take the first and second derivatives of x(t) with respect to time.

By applying the product rule and chain rule, we can obtain the following:

dx/dt = A(k1)e^(k1t) + B(k2)e^(k2t)

d^2x/dt^2 = A(k1)^2e^(k1t) + B(k2)^2e^(k2t)

Now, we can substitute these expressions into the given differential equation and simplify:

(d^2x/dt^2) + 2B(dx/dt) + (w^2)x = (A(k1)^2e^(k1t) + B(k2)^2e^(k2t)) + 2B(A(k1)e^(k1t) + B(k2)e^(k2t)) + (w^2)(Ae^(k1t) + Be^(k2t))

= (A(k1)^2 + 2BA(k1) + w^2A)e^(k1t) + (B(k2)^2 + 2BB(k2) + w^2B)e^(k2t)

= (k1^2 + 2Bk1 + w^2)Ae^(k1t) + (k2^2 + 2Bk2 + w^2)Be^(k2t)

Since k1 and k2 are the distinct roots of the equation k^2 + 2Bk + w^2, we can simplify further to obtain:

(d^2x/dt^2) + 2B(dx/dt) + (w^2)x = 0

Which is the same as the original differential equation. Therefore, we have proven that x(t) = Ae^(k