Solution Passing Through Point

Question

If dy/dx = -sin^2 y, then the solution passing through point (π/4, π/2) is

A. y = cot^-1 (x - π/4)

B. y = cot^-1 (-x + π/4)

C. y = cot^-1 (x - π/2 + 1)

D. y = cot^-1 (-x + π/2 + 1)

Answer 1

Hi Lissie,

Rearrange the differential equation such that

-dy / (sin²(y)) = dx. Recall that 1 / sin(y) = csc(y), so the differential equation becomes

-csc²(y) dy = dx.

The integral of -csc²(x) is cot(x) + C, so integrating both sides of the differential equation yields cot(y) = x + C.

Substituting the initial condition (π/4, π/2) for x and y, respectively, yields cot(π/2) = π/4 + C. Since cot(π/2) = 0, you have that C = 0 - π/4 = -π/4. Substituting the value of C and solving for y in the original equation gives the answer of

y = cot(x - π/4) ⇒ A

Hope this helps!

Akshat Y.