The general solution of the differential equation (1 + x^2)displaysty…

The correct answer is Option (4) → \(y(1 + x^2) = \displaystyle\frac{4x^3}{3} + C\)

Dividing by \(1 + x^2\) gives a linear equation:

    \( \displaystyle \frac{dy}{dx} + \frac{2x}{1 + x^2}\,y = \frac{4x^2}{1 + x^2} \)

The integrating factor is

    \( \displaystyle e^{\int \frac{2x}{1 + x^2}dx} = e^{\log(1 + x^2)} = 1 + x^2 \)

So

    \( \displaystyle y(1 + x^2) = \int \frac{4x^2}{1 + x^2}\cdot(1 + x^2)\,dx = \int 4x^2\,dx \)

    \( \displaystyle y(1 + x^2) = \frac{4x^3}{3} + C \)

In fact the left side of the given equation is already \(\frac{d}{dx}\left[(1 + x^2)y\right]\), which gives the same result at once.