Q98 / 150 · MHT CET 2026 (20 April Shift 1) · Mathematics
If $y = f(x)$ is a monotonically increasing function such that $\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx}$ and $y(0) = 5$, then $y(3) = \cdots$
- A$23$
- B$14$
- C$13$
- Correct answerD$11$
Solution
Let $\dfrac{dy}{dx} = p$
The given differential equation is $p^2 + p - 6 = 0$
$(p+3)(p-2) = 0$
$p = -3$ or $p = 2$
Since $y = f(x)$ is a monotonically increasing function, $\dfrac{dy}{dx} > 0$
Therefore, $\dfrac{dy}{dx} = 2$
Integrating both sides, we get $y = 2x + C$
Given $y(0) = 5$, we have $5 = 2(0) + C \Rightarrow C = 5$
Thus, $y = 2x + 5$
Substituting $x = 3$, we get $y(3) = 2(3) + 5 = 11$
Answer: $11$
The given differential equation is $p^2 + p - 6 = 0$
$(p+3)(p-2) = 0$
$p = -3$ or $p = 2$
Since $y = f(x)$ is a monotonically increasing function, $\dfrac{dy}{dx} > 0$
Therefore, $\dfrac{dy}{dx} = 2$
Integrating both sides, we get $y = 2x + C$
Given $y(0) = 5$, we have $5 = 2(0) + C \Rightarrow C = 5$
Thus, $y = 2x + 5$
Substituting $x = 3$, we get $y(3) = 2(3) + 5 = 11$
Answer: $11$
Hence, the correct answer is option: 'D'.