solve the differential equation ds/d(theta) = (2csc^2 theta)/ (cot theta). Then find the particular solution at (pi/4, 4).

Question

solve the differential equation ds/d(theta) = (2csc^2 theta)/ (cot theta). Then find the particular solution at (pi/4, 4).

Answer 1

ds/dθ = 2csc²(θ)/cot(θ)

∫ds = ∫2csc²(θ)/cot(θ) dθ

let u = cot(θ), du = -csc²(θ)dθ, dθ= -du/csc²(θ)

s = -2∫du/u = -2ln|u| + C = -2ln|cot(θ)| + C

s(θ) = -2ln|cot(θ)| + C

s(π/4) = -2ln(1) + C = 4 --> C= 4

s(θ) = -2ln|cot(θ)| + 4