Let y=y(x) be the solution of the differential equation x ~d y ~d x -y=x^ 2 x, x ∈(0, π) . If y((π)/(2))=(π)/(2) , then 6 y((π)/(6))-8 y((π)/(4)) is equal to :

Correct answer: C. -π The differential equation x(dy)/(dx) - y = x^2 x can be rewritten as (dy)/(dx) - (y)/(x) = x x . The integrating factor is μ(x) = (1)/(x) . Multiplying by μ(x) gives (d)/(dx)((y)/(x)) = x . Integrating: (y)/(x) = | x| + C , so y = x ( x) + Cx . Using y((π)/(2)) = (π)/(2) : we get (π)/(2) = 0 + C(π)/(2) , thus C = 1 . Therefore y = x(1 + ( x)) . Now: y((π)/(6)) = (π)/(6)(1 - 2) and y((π)/(4)) = (π)/(4)(1 - ( 2)/(2)) . Thus 6y((π)/(6)) - 8y((π)/(4)) = π(1 - 2) - 2π(1 - ( 2)/(2)) = -π .

JEE Main 202628 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 28 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Differential Equations

Let y=y(x) be the solution of the differential equation x ~d y ~d x...

Let

y=y(x)

be the solution of the differential equation

x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y=x^{2} \cot x, x \in(0, \pi)

. If

y\left(\frac{\pi}{2}\right)=\frac{\pi}{2}

, then

6 y\left(\frac{\pi}{6}\right)-8 y\left(\frac{\pi}{4}\right)

is equal to :

Options

A $3 \pi$
B $\pi$
C $-\pi$
D $-3 \pi$

Correct answer

C. $-\pi$

Step-by-step solution

The differential equation

x\frac{dy}{dx} - y = x^2 \cot x

can be rewritten as

\frac{dy}{dx} - \frac{y}{x} = x \cot x

.
The integrating factor is

\mu(x) = \frac{1}{x}

. Multiplying by

\mu(x)

gives

\frac{d}{dx}\left(\frac{y}{x}\right) = \cot x

.
Integrating:

\frac{y}{x} = \ln|\sin x| + C

, so

y = x\ln(\sin x) + Cx

.
Using

y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}

: we get

\frac{\pi}{2} = 0 + C\frac{\pi}{2}

, thus

C = 1

.
Therefore

y = x(1 + \ln(\sin x))

.
Now:

y\left(\frac{\pi}{6}\right) = \frac{\pi}{6}(1 - \ln 2)

and

y\left(\frac{\pi}{4}\right) = \frac{\pi}{4}\left(1 - \frac{\ln 2}{2}\right)

.
Thus

6y\left(\frac{\pi}{6}\right) - 8y\left(\frac{\pi}{4}\right) = \pi(1 - \ln 2) - 2\pi\left(1 - \frac{\ln 2}{2}\right) = -\pi

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