Question

Let $C_1$ and $C_2$ be two curves which satisfy the differential equation $2{\left(\frac{dy}{dx}\right)}^2$ + $x\left(\frac{dy}{dx}\right)=y$ and passes through (1, 1). If the area enclosed by the curves $C_1$, $C_2$ and the axes is $\frac{m}{R}$ then m + n = _______________

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Answer 1

$\frac{dy}{dx}=\frac{-x\pm \sqrt{x^2+8y}}{4}\Rightarrow x=2y-1$ and $x+y=2$

$\therefore Area=\frac{5}{4}=\frac{m}{n}$