Let y = y(x) be the solution of the differential equation x dy/dx − y = x² cot x

Q. Let y = y(x) be the solution of the differential equation x dy/dx − y = x² cot x, x ∈ (0, π). If y(π/2) = π/2, then 6y(π/6) − 8y(π/4) is equal to :

A. −3π

B. 3π

C. −π

D. π

Correct Answer: −π

Explanation

The given differential equation is:

x dy/dx − y = x² cot x

Dividing throughout by x:

dy/dx − (1/x)y = x cot x

This is a linear differential equation of the form dy/dx + Py = Q, where P = −1/x.

Integrating factor (I.F.) = e^{∫(−1/x)dx} = 1/x

Multiplying the equation by 1/x:

d/dx (y/x) = cot x

Integrating both sides:

y/x = ln(sin x) + C

So,

y = x ln(sin x) + Cx

Using the given condition y(π/2) = π/2:

π/2 = (π/2) ln(1) + C(π/2)

⇒ C = 1

Thus,

y = x ln(sin x) + x

Now,

y(π/6) = (π/6)[ln(1/2) + 1]

y(π/4) = (π/4)[ln(1/√2) + 1]

Substituting in the given expression:

6y(π/6) − 8y(π/4) = −π

Hence, the required value is −π.

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