Interpret the initial value problem

Question

Interpret the initial value problem

d²θ/dt²+ß²θ=0, θ(0)=θ₀ , dθ/dt]_t=0=ω₀

for any physical situation and hence solve the problem.

Answer 1

The goal here is to find θ(t).

Your equation is second order homogenous.

I. Find the roots

It becomes: r² + ß² = 0

r = √-ß = ß√-1

r = 0 + ßi (I place zero there because r = λ ± µi is the form which completes the solution)

λ = 0

µ = ß

∴ We get eq. (1) θ(t) = Asinßt + Bcosßt

II. Initial Conditions to find Coefficients

θ(t=0) = θ₀ = Asin(0) + Bcos(0)

θ₀ = B

(1) becomes

(2) θ(t) = Asinßt + θ₀cosßt

dθ_t = Aßcosßt - θ₀ßsinßt

dθ_t (t=0) = ω₀ = Aßcos(0) - θ₀ßsin(0)

A = ω₀⁄ß

(2) becomes

(3) θ(t) = (ω₀⁄ß)sinßt + θ₀cosßt

(3) Is your solution.

Answer 2

To make things easier to read I am going to use "prime" notation for derivatives, i.e. instead of writing

d²θ/dθ²

I'll write

θ''

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

θ'' = -β²θ

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

θ = sin(t) or θ = cos(t)

at least in the case where β = 1. Considering it for a while more, we can get that factor of β² there by letting

θ = sin(βt) or θ = cos(βt)

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

θ = a*sin(βt) + b*cos(βt)

Plugging in t=0 and using our first initial value, we get that b = θ₀. If we differentiate the above equation and plug in our second initial value, we get a = ω₀/β. Putting this together, we have

θ₀ = (ω₀/β)*sin(βt) + (θ₀)*cos(βt)

You can go further if you want, using one of the angle sum or difference 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 acceleration there in the differential equation when we see θ''. 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 θ'' in the differential equation we could replace it with F/m. On the other side we just have displacement times a negative constant. So this is

F/m = -β²θ

Renaming constants this is

F = -cθ

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 the phase angle determined by the initial conditions. Getting more precise with this will get us the same answer we got in the first part.