Given $y=f(x)$ , is twice differentiable, passes through the origin and satisfies the equation, $$\frac{dy}{dx} +\int_0^5{y\,dx}=27$$What is the probability that $2$ randomly chosen variables $a$, $b$ from the set $S=\{1,2,3,4\}$ lies on the curve as $(a,b)$?
My Attempt:
As $\int_0^5{y\,dx}$ is a constant.
$$\frac{d^2y}{dx^2}=0$$
Therefore, $y=ax+b$,
As curve passes through the origin, $b=0$, so $y=ax$.
On putting this in the equation, we get $a=2$.
Therefore, $$y=2x.$$
Is my approach right? And also how else can we attempt this question?
You have $y=2x$ so the only integer combinations from your set $S$ are $(1, 2)$ and $(2, 4)$.
The probability is thus $\dfrac2{4\cdot4}=\dfrac18$.