Find the particular solution of the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \quad \text{given that} \quad y = \frac{\pi}{4}, \text{ when } x = 1. \]

Question

Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \] is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \] find the values of $\alpha$ and $\beta$.

Answer 1

The differential equation $ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 $ is provided. This equation can be rewritten as $ \sin^2 \left( \frac{y}{x} \right) \, dx + \, dy = 0 $. To solve this by separating variables, we first divide by $ x $, yielding $ \sin^2 \left( \frac{y}{x} \right) \, dx = -dy $. Integrating both sides, we get $ \int \sin^2 \left( \frac{y}{x} \right) \, dx = \int -dy $. The solution is obtained through substitution and simplification using standard integration methods. The particular solution, satisfying the initial condition $ y = \frac{\pi}{4} $ at $ x = 1 $, is the result of this integration.

Find the particular solution of the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \quad \text{given that} \quad y = \frac{\pi}{4}, \text{ when } x = 1. \]

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