Question

Solve the differential equation: \[ x \frac{dy}{dx} - y + x \sin \left(\frac{y}{x}\right) = 0. \]

Hint:

Use the substitution \( v = \frac{y}{x} \) in equations of the form: \[ x \frac{dy}{dx} - y = f(y/x). \]

Answer 1

Step 1: Use Substitution
Let \( v = \frac{y}{x} \), so that: \[ y = vx \quad {and} \quad \frac{dy}{dx} = v + x \frac{dv}{dx}. \] Step 2: Rewrite the Equation
\[ x (v + x \frac{dv}{dx}) - vx + x \sin v = 0. \] \[ x v + x^2 \frac{dv}{dx} - vx + x \sin v = 0. \] \[ x^2 \frac{dv}{dx} = -x \sin v. \] Step 3: Solve the Separable Equation
\[ \frac{dv}{\sin v} = -\frac{dx}{x}. \] Step 4: Integrate Both Sides
\[ \int \frac{dv}{\sin v} = -\int \frac{dx}{x}. \] \[ \log |\csc v - \cot v| = -\log |x| + C. \] Step 5: Substitute Back \( v = \frac{y}{x} \)
\[ \log |\csc (y/x) - \cot (y/x)| = -\log |x| + C. \] Taking exponentials on both sides: \[ |\csc (y/x) - \cot (y/x)| = \frac{C}{x}. \]