\( ( 1 + x ^ { 2 } ) + ( y ^ { 2 } + x ^ { 2 } y ^ { 2 } ) d y = ( 11 y ^ { 2 } ) d x \)

To solve the given differential equation \( ( 1 + x ^ { 2 } ) + ( y ^ { 2 } + x ^ { 2 } y ^ { 2 } ) d y = ( 11 y ^ { 2 } ) d x \), we can rearrange it into a more manageable form. By isolating \( dy \) and \( dx \), we can express it as: \[ \frac{dy}{dx} = \frac{11y^2 - (1 + x^2)}{y^2 + x^2y^2} \] This format allows us to identify potential techniques for solving such equations, such as separation of variables or finding an integrating factor depending on whether it's exact or not. Remember, the key to solving first-order differential equations efficiently often lies in recognizing the structure of the equation early on. Additionally, exploring the properties of similar types of equations can yield insights. For instance, equations that include both \(y\) and \(x\) often allow substitutions that can simplify them into separable forms. Don't underestimate polynomial factorization—sometimes it can unlock a solution hiding in plain sight!