Answer to Question #102899 in Differential Equations for Ajay

Question

Interpret the initial value problem d^2θ/dt^2+ß^2 θ=0, θ(0)=θ° ,dθ/dt] at (t=0) =ω°
for any physical situation and hence solve the problem

Answer 1

To use "prime" notation for derivatives,i.e instead of writing

$\dfrac{d^2θ }{dθ^2}$

I will write

$\theta''$

Now without the initial condition, and by moving the second term across the equals sign, our ordinary differential equation is

$\theta''=-\beta^2\theta$

Think about the functions whose second derivative is a negative multiple of the function itself. If you consider a few elementary functions eventually you stumble upon.

$\theta=sin(t)$ or $\theta$ $=cos(t)$

at least in the case where $\beta=1$ .Considering it for a while more,we can get that factor of$\beta^2$ thereby letting

$\theta=sin(βt)$ or $\theta$ $=cos(βt)$

In fact any linear combination of these will do,so could have

$\theta=a*sin(βt)+b*cos(βt)$

plugging t=0 and using our first initial value,we get that b=$\theta_0$ .If we differentiate the above equation and plug in our second initial value,we get a=$\omega_0/\beta$ .Putting this together,we have

$\theta_0=(\omega_0/\beta)*sin(βt)+(\theta_0)*cos(βt)$

You can go further if you want ,using one of the angle sum or differences formulas to write this as a single sine or cosine wave,but right now this is probably enough.

Of course ,we were asked to interpret this physically .We are seeing this acceleration there in the differential equation when we see $\theta''$ .Acceleration also makes an appearance in Newton's second law:Force=mass times acceleration.F=ma.We could write this as a=F/m,in which case when we see $\theta''$

in the differential equation we could replace it with F/m.On the other side we just have displacement times have a negative constant.So this is

F/m=$-\beta^2\theta$

Remaining constants this is

F=-c$\theta$

That is, force is proportional to displacement and in the opposite direction of displacement.This is how

a spring mass system is modeled when damping forces are ignored.Which is a cosine wave,with the amplitude and phase angle determined by the initial conditions.Getting more precise with this will get us the same answer we got in the first part.