Question

Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, (x \neq 0) \] given that \( y = 0 \) if \( x = \frac{\pi}{2} \).

Hint:

For linear first-order differential equations, use the integrating factor method to simplify the equation and then integrate.

Answer 1

The given differential equation is: \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \] This is a linear first-order differential equation of the form: \[ \frac{dy}{dx} + P(x)y = Q(x) \] Where \( P(x) = \cot x \) and \( Q(x) = 2x + x^2 \cot x \). The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \cot x \, dx} = e^{\ln \sin x} = \sin x \] Multiply both sides of the differential equation by \( \mu(x) = \sin x \): \[ \sin x \frac{dy}{dx} + y \sin x \cot x = (2x + x^2 \cot x) \sin x \] This simplifies to: \[ \frac{d}{dx} \left( y \sin x \right) = 2x \sin x + x^2 \cos x \] Now, integrate both sides: \[ \int \frac{d}{dx} \left( y \sin x \right) \, dx = \int \left( 2x \sin x + x^2 \cos x \right) \, dx \] The left side is: \[ y \sin x \] To integrate the right-hand side, break it into two integrals: \[ \int 2x \sin x \, dx \text{and} \int x^2 \cos x \, dx \] Use integration by parts for both terms. For \( \int 2x \sin x \, dx \), we get: \[ \int 2x \sin x \, dx = -2x \cos x + 2 \sin x \] For \( \int x^2 \cos x \, dx \), use integration by parts again: \[ \int x^2 \cos x \, dx = x^2 \sin x - 2x \cos x + 2 \sin x \] Thus, the right-hand side becomes: \[ -2x \cos x + 2 \sin x + x^2 \sin x - 2x \cos x + 2 \sin x = -4x \cos x + x^2 \sin x + 4 \sin x \] Thus, the general solution is: \[ y \sin x = -4x \cos x + x^2 \sin x + 4 \sin x + C \] Now, substitute \( x = \frac{\pi}{2} \) and \( y = 0 \): \[ 0 \cdot 1 = -4 \cdot \frac{\pi}{2} \cdot 0 + \left( \frac{\pi}{2} \right)^2 \cdot 1 + 4 \cdot 1 + C \] Solving for \( C \): \[ C = -4 + \frac{\pi^2}{4} + 4 = \frac{\pi^2}{4} \] Thus, the particular solution is: \[ y \sin x = -4x \cos x + x^2 \sin x + 4 \sin x + \frac{\pi^2}{4} \] \[ y = \frac{-4x \cos x + x^2 \sin x + 4 \sin x + \frac{\pi^2}{4}}{\sin x} \] Final Answer: \[ \boxed{y = \frac{-4x \cos x + x^2 \sin x + 4 \sin x + \frac{\pi^2}{4}}{\sin x}} \]