How do you find the particular solution to #ysqrt(1-x^2)y'-x(1+y^2)=0# that satisfies y(0)=sqrt3?
Use the separation of variables method, integrate both sides, and then use the specified point to evaluate the constant.
Use the notation
Move the second term to the right side:
Multiply both sides of the equation by
Integrate both sides:
Multiply both sides by 2:
Use the exponential function:
Adding a constant in the exponent is the same as multiplying by a constant:
Subtract 1 from both sides:
Square root both sides:
Use the point to solve for C, substitute 0 for x and
The above equation can only be true, if we can drop the
Square both sides and solve for C: