Solve the differential equation dy dx equals the quotient of y times the cosine of x and the quantity 1 plus y squared with the initial condition y(0) = 1

Question

Solve the differential equation  with the initial condition y(0) = 1

Answer 1

Multiply both sides by "1+y²", divide both sides by "y", multiply both sides by "dx" and you get:

(1 + y²)/y dy = cos(x) dx

Integrate both sides. The left side can be simplified as 1/y + y

∫(1/y + y) dy = ∫cos(x) dx

lny + y²/2 = sin(x) + C

Plug in the initial conditions (0, 1) to get C = 1/2

lny + y²/2 = sin(x) + 1/2