If differential equation of the curve $x^2 - y^2 = c\left(x^2 + y^2\right)^2$ is $\frac{dy}{dx} = \frac{x(my^2 - x^2)}{y(3x^2 + my^2)}$, then $(m + n)$ is ____.
Step-by-Step Solution
Key Concept: Differentiate the implicit curve equation $x^2 - y^2 = c(x^2 + y^2)^2$ using implicit differentiation to extract the differential equation form, then match coefficients with $ rac{dy}{dx} = rac{x(my^2 - x^2)}{y(3x^2 + my^2)}$ to identify $m$ and $n$.
Given $\frac{dy}{dx} = \frac{x(3y^2 - x^2)}{y(3x^2 - y^2)}$, this is a homogeneous equation. Using the substitution $y = vx$, we transform it to separate variables in $v$ and $x$. After simplification and integration, the solution yields $m = 3$ and $n = -1$.
Correct Answer: 2