Solving a differential equation for oscillatory mechanics

Question

Well I was studying oscillatory mechanics and I got stuck on an differential equation which I am not able to solve.

The equation is $$ ma= ( -kx) + ( -bv) + F\sin(¥t+\Delta) $$ where $a$ is the acceleration of the particle, $x$ is the displacement from the mean position, $F$ is the applied force, $¥$ is the frequency of the applied force $\Delta$ is the phase constant, and $m$ is the mass of the particle.

I want to solve this equation for displacement. I don't know how to solve a 2nd order differential equation.Please explain me as I am a high school kid. Please provide the solution to this equation.

Answer 1

Assuming $b$ and $k$ are also constants, you have a 2nd order ODE with constant coefficients for $x = x(t)$: $$ mx''(t) + bx'(t)+kx(t)=F\sin(Yt+\Delta) $$ As is usual in such situations, the approach is as follows:

1. Solve the homogeneous equation in general form, assuming the form of the solution to be $x_h(x) = e^{ax}$, you should get 2 independent families to solve LHS = 0.
2. Pick a particular solution $x_p(t)$ by a variety of techniques, e.g. variation of parameters, or just observation, here you are likely to get solutions of the form $$x_p(t) = A\sin(Yt+\Delta) + B\cos(Yt+\Delta)$$
3. Your final solution will be of the form $x(t) = x_h(t) + x_p(t)$.