Verified answer

That is a separable first order differential equation [1].

To solve it separate the variables first:

θ (dr/dθ) = - 2∙r

<=>

(1/r) dr = - (2/θ) dθ

Next step is to integrate each side:

∫ (1/r) dr = ∫ - (2/θ) dθ

=>

ln(r) = - 2∙ln(θ) + c

(c is the constant of integration)

Using the logarithmic identity

ln(xⁿ) = n∙ln(x)

the solution can be rewritten as

ln(r) = c + ln(θ⁻²) = c + ln(1/θ²)

Hence,

r = e^(c + ln(1/θ²)) = e^(c)∙ln(1//θ²) = e^(c)/θ²

By setting e^(c) you get the general solution:

r = C/θ²

The constant C can be found by applying the initial condition

at θ = -1/3

r = 9

=>

9 = C / (-1/3)²

<=>

9 = 9∙C

<=>

C = 1

So the solution to this initial value problem is:

r = 1/θ²