Differential Equations

Identify the form: This is a Linear Differential Equation of the form $d y / d x + P (x) y = Q (x)$ . $P (x) = 1/ x, Q (x) = x^{2}$ .
Find Integrating Factor (IF): IF $= e^{\int (1/ x) d x} = e^{l n x} = x$ .
Solution formula: $y \cdot IF = \int (Q \cdot IF) d x + C$ $x y = \int (x^{2} \cdot x) d x + C$ $x y = \int x^{3} d x + C$ $x y = x^{4} /4 + C$
Isolate $y$ : $y = x^{3} /4 + C / x$ .

This easy difficulty mathematics question is from the chapter differential equations, covering the topic of differential equations. It appeared in the 2025 exam.

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Differential Equations

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Solution

Let $f (x)$ be differentiable s.t. $f^{'} (x) = 2 - f (x) / x$ and $f (1) = 1$ . Then $f (x)$ is:

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