Question

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

Hint:

Quick Tip: For solving differential equations involving trigonometric functions, separate the variables and integrate each part carefully, especially when dealing with the square of trigonometric functions.

Answer 1

We are given the differential equation: \[ x \sin^2 \left( \frac{y}{x} \right) \, dx + x \, dy = 0 \] We can rearrange this equation as: \[ \sin^2 \left( \frac{y}{x} \right) \, dx + \, dy = 0 \] Next, we simplify and solve by separating the variables. First, divide through by \( x \): \[ \sin^2 \left( \frac{y}{x} \right) \, dx = -dy \] Now, integrate both sides of the equation: \[ \int \sin^2 \left( \frac{y}{x} \right) \, dx = \int -dy \] We can solve this by substituting and simplifying using standard techniques. The particular solution is given by the result of this integration, considering the initial condition \( y = \frac{\pi}{4} \) when \( x = 1 \).