Derive the equation of damped simple harmonic motion and solve differential equation

# Doubt from Damped simple harmonic motion

**Tanmay_1**#1

You need complex analysis to solve such an equation nicely.

The differential equation will be of the form : \frac{md^2x}{dt^2}+\gamma\frac{dx}{dt}+m\omega_0^2x=0 , where \omega_0 is the natural frequency of oscillation.\space\space\space(1).

Now, this is a second ordered differential equation and for solving it, we consider another equation in a variable y.

\implies \frac{md^2y}{dt^2}+\gamma\frac{dy}{dt}+m\omega_0^2y=0\space\space\space(2)

From (1)+i(2), we get : \frac{md^2z}{dt^2}+\gamma\frac{dz}{dt}+m\omega_0^2z=0\space\space\space(3)

where z=x+iy is any complex number.

Now, we consider a trial solution of the form : z(t)=z_0e^{rt} and put this in (3).

On putting the value of z(t) we will get a quadratic equation in r, which can be solved for two values of r and the general solution will then be a superposition of the two solutions.

Do let me know if you have any doubt.

**Tanmay_1**#3

Sir ...i know the answer but uploaded the question for others to familarise ..i am 49 year old guy...

Sir, I have already told you, this is a dedicated forum mainly for JEE and BITSAT so kindly post questions or so that are related to that. If we are loading the aspirants with these kinda questions and approaches, they will be deviated from their JEE studies and that's what we are here to prevent them from.

**Tanmay_1**#5

Thank you sir...i shall follow instruction...if possible please delete my questions which are not related with JEE syllabus...