Solve the Separable equation
y=((sqrt(1-y^2)) x^2
To solve the separable equation y = sqrt(1 - y^2) * x^2, we can follow these steps:
Step 1: Rewrite the equation in a standard form to separate the variables.
Divide both sides by x^2:
y / x^2 = sqrt(1 - y^2)
Step 2: Rewrite the square root on the right-hand side (RHS) to get rid of the radical.
Square both sides to eliminate the square root:
(y / x^2)^2 = (sqrt(1 - y^2))^2
y^2 / x^4 = 1 - y^2
Step 3: Multiply both sides by x^4 to eliminate denominators and bring all terms to one side.
y^2 = x^4 - x^4y^2
Step 4: Rearrange the equation to have all terms containing y on one side and all terms without y on the other side.
x^4y^2 + y^2 = x^4
y^2(x^4 + 1) = x^4
Step 5: Divide both sides by (x^4 + 1) to isolate y^2.
y^2 = x^4 / (x^4 + 1)
Step 6: Take the square root of both sides to solve for y.
y = sqrt(x^4 / (x^4 + 1))
Thus, the solution to the separable equation y = sqrt(1 - y^2) * x^2 is y = sqrt(x^4 / (x^4 + 1)).