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.

We rearrange the given differential equation to combine all the y terms on the left side of the equation and the x terms on the right side:

[ (1 + y^2) / y] dy = cos (x) dx

[ (1/y) + y] dy = cos (x) dx

We integrate both sides to get

ln y + (y^2 / 2) = sin (x) + C

Then, we apply the initial condition y = 1 at x = 0 to get the value of C:

ln (1) + (1^2 / 2) = sin (0) + C

C = ln (1) + (1^2 / 2) - sin (0)

C = 1/2

Therefore, our final answer is

ln (y) + (y^2 / 2) = sin (x) + 1/2