JEE Main 2025 Jan 23 Shift 2, Mathematics Q11: Methods of Solving First Order, First Degree Differential Equation

JEE Main2025Jan 23, Shift 2Mathematics

Let $x = x (y)$ be the solution of the differential equation

$y = (x - y \frac{d x}{d y}) sin (\frac{x}{y}), y > 0, x (1) = \frac{π}{2} .$

Then $cos (x (2))$ is equal to :

Solution

Rewrite to expose an exact differential. The right side contains $x d y - y d x$ when both sides are divided by $d y$ :

$y d y = (x d y - y d x) sin (\frac{x}{y}) .$

Divide by $y^{2}$ : $\frac{d y}{y} = \frac{x d y - y d x}{y ^{2}} sin (\frac{x}{y}) .$

Notice $d (\frac{x}{y}) = \frac{y d x - x d y}{y ^{2}}$ , so the right side is $- sin (\frac{x}{y}) d (\frac{x}{y}) = d (cos \frac{x}{y}) .$

Integrate: $ln y = cos (\frac{x}{y}) + C .$

Apply $x (1) = \frac{π}{2}$ : $ln 1 = cos (\frac{π}{2}) + C \Rightarrow 0 = 0 + C \Rightarrow C = 0.$

At $y = 2$ : $ln 2 = cos (\frac{x}{2})$ .

Use the double-angle identity $cos x = 2 cos^{2} (\frac{x}{2}) - 1 = 2 (ln 2)^{2} - 1.$

Concept behind this question

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