Solution:

Given equation: (1-x^2)dy - xydx = xy^2dx

To solve this equation, we can use the method of exact differential equations.

Step 1: Check for Exactness
To check if the equation is exact, we need to verify if the following condition is satisfied:

∂M/∂y = ∂N/∂x

Where M and N are the coefficients of dy and dx respectively.

In this case, M = (1-x^2) and N = -xy^2

Taking the partial derivatives, we have:

∂M/∂y = 0
∂N/∂x = -y^2

Since ∂M/∂y is not equal to ∂N/∂x, the equation is not exact.

Step 2: Multiply by an Integrating Factor
To make the equation exact, we can multiply both sides by an integrating factor, which is an expression involving only x or y.

In this case, we can choose the integrating factor as μ = 1/(xy^3). Multiplying both sides of the equation by μ, we get:

μ(1-x^2)dy - μxydx = μxy^2dx

Simplifying, we have:

(1-x^2)/y^3 dy - x/y^2 dx = xdx

Step 3: Check for Exactness Again
Now, let's check if the equation is exact after multiplying by the integrating factor.

∂M/∂y = (1-x^2)/y^3
∂N/∂x = -x/y^2

Taking the partial derivatives, we have:

∂M/∂y = (1-x^2)/y^3
∂N/∂x = -x/y^2

Since ∂M/∂y is equal to ∂N/∂x, the equation is now exact.

Step 4: Find the General Solution
To find the general solution, we integrate each term with respect to its variable.

Integrating (1-x^2)/y^3 dy, we get:

∫ (1-x^2)/y^3 dy = ∫ xdx

Simplifying and integrating, we have:

-1/(2y^2) + (x^2/2y^2) = (x^2/2) + C

Where C is the constant of integration.

Step 5: Simplify the Solution
To simplify the solution, we can multiply both sides of the equation by 2y^2:

-1 + x^2 = y^2(x^2 + C)

Rearranging the terms, we have:

x^2 + C = (y^2 - 1)(x^2)

Expanding the right side of the equation, we get:

x^2 + C = x^2y^2 - y^2

Finally, rearranging the terms, we have:

(x^2 + C) - x^2y^2 + y^2 = 0

(x^2 - x^2y^2 + y^2) + C = 0

(x^2(1 - y^2) + y^2) + C = 0

(x^2(1 - y