Find an explicit solution of the given initialvalue problem

Find an explicit solution of the given initial-value problem.

Find an explicit solution of the given initial-value problem. root 1-y^2 dx - root 1-x^2 dy = 0,y (0), y (0) = root 2/2

Solution

Rewriting :

dy/srt(1 - y^2) = dx/sqrt(1 - x^2)

Integrating :

arcsin(y) = arcsin(x) + C

y = sin[arcsin(x) + C]

y(0) = sqrt2/2 :

sqrt2/2 = sin[arcsin(0) + C]

arcsin(sqrt2/2) = 0 + C

pi/4 = C

So, solution, y = sin[arcsin(x) + C]
becomes

y = sin(arcsin(x) + pi/4) ---> ANSWER