How can we solve the differential equation \\( rac{dy}{dx} = x(1 – y^2) \\) using separation of variables?

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To solve the differential equation \\( rac{dy}{dx} = x(1 – y^2) \\) using separation of variables, we follow these steps:

  1. Separate the variables: We want to rearrange the equation so that all terms involving \\( y \\) are on one side and all terms involving \\( x \\) are on the other side. Starting with the given equation:

    2. \\[ rac{dy}{dx} = x(1 – y^2) \\]

    We can rewrite this as:

    \\[ rac{1}{1 – y^2} dy = x \, dx \\]

  2. Integrate both sides: Now we integrate both sides. The left side involves a simple integral, while the right side is straightforward:

    3. Integrating the left side, we have:

    \\[ ext{Let } u = 1 – y^2, ext{ then } du = -2y \, dy \\]

    Thus:

    \\[ - rac{1}{2} ext{ln} |1 – y^2| = rac{x^2}{2} + C \\]

  3. Combine terms and isolate \\(y \\: :

    4. Now, we solve for \\( y \\). First, we rearrange the equation from the integral step:

    \\[ ext{ln} |1 – y^2| = -x^2 + C’ \\]

  4. Exponentiate both sides:

    5. Removing the logarithm by exponentiating gives us:

    \\[ |1 – y^2| = e^{-x^2 + C’} = C e^{-x^2} \\]

  5. Final solution: Solving for \\( y \\: We take the positive case, but also consider the negative:

    6. \\[ 1 – y^2 = Ce^{-x^2} \\]

    Which rearranges to:

    \\[ y^2 = 1 – Ce^{-x^2} \\]

    Thus, isolating \\(y\:

    \\[ y = ext{±} \, ext{sqrt}(1 – Ce^{-x^2}) \\

In summary, we have solved the differential equation through separation of variables, leading us to the solution:

\\[ y = ext{±} \, ext{sqrt}(1 – Ce^{-x^2}) \\[

Where \\( C \\) is a constant determined by initial conditions, if provided.

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