JEE Math Practice Question

Let $$f(x)$$ be a positive function such that the area bounded by $$y=f(x), y=0$$ from $$x=0$$ to $$x=a>0$$ is $$e^{-a}+4 a^2+a-1$$. Then the differential equation, whose general solution is $$y=c_1 f(x)+c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is

1. A.$$\left(8 e^x+1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$$
2. B.$$\left(8 e^x+1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$$
3. C.$$\left(8 e^x-1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$$
4. D.$$\left(8 e^x-1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$$