JEE Main 2025 Apr 2 Shift 2, Mathematics Q2: Methods of Solving First Order, First Degree Differential Equation

JEE Main2025Apr 2, Shift 2Mathematics

Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $10 \int_{1}^{x} f (t) d t = 5 x f (x) - x^{5} - 9$ for all $x \geq 1$ , then the value of $f (3)$ is :

A
$18$
B
$32$
C
$22$
D
$26$

Solution

Differentiate both sides with respect to $x$ :

$10 f (x) = 5 f (x) + 5 x f^{'} (x) - 5 x^{4} \Rightarrow x f^{'} (x) - f (x) = x^{4}$ .

Dividing by $x$ , $\frac{d y}{d x} - \frac{y}{x} = x^{3}$ , a linear ODE with integrating factor $\frac{1}{x}$ :

$\frac{d}{d x} (\frac{y}{x}) = x^{2} \Rightarrow \frac{y}{x} = \frac{x ^{3}}{3} + C$ .

Determine $C$ using $x = 1$ in the original integral identity: $0 = 5 f (1) - 1 - 9 \Rightarrow f (1) = 2$ . So $\frac{2}{1} = \frac{1}{3} + C \Rightarrow C = \frac{5}{3}$ .

Therefore $f (x) = \frac{x ^{4}}{3} + \frac{5 x}{3}$ , giving $f (3) = \frac{81 + 15}{3} = 32$ .

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JEE Main 2025 Apr 2 Shift 2, Mathematics Q2: Methods of Solving First Order, First Degree Differential Equation

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