Question

Solve the following differential equation : \( x \frac{dy}{dx} = y - x \sin^2 \left( \frac{y}{x} \right) \), given that y(1) = \( \frac{\pi}{6} \).

Hint:

The presence of \( y/x \) inside a function is a clear hint to use homogeneous substitution.

Answer 1

Concept:
• Homogeneous differential equation: Substitutions \( y = vx \).
• Variable separable form.

Step 1: Convert to homogeneous form and substitute
Divide by \( x \): \( \frac{dy}{dx} = \frac{y}{x} - \sin^2\left(\frac{y}{x}\right) \).
Let \( y = vx \), then \( \frac{dy}{dx} = v + x \frac{dv}{dx} \).
\[ v + x \frac{dv}{dx} = v - \sin^2 v \implies x \frac{dv}{dx} = -\sin^2 v \]

Step 2: Separate variables and integrate
\[ \frac{dv}{\sin^2 v} = -\frac{dx}{x} \implies \int \csc^2 v dv = -\int \frac{1}{x} dx \]
\[ -\cot v = -\log |x| + C \implies \cot(y/x) = \log |x| + C \]

Step 3: Solve for C
Using \( y(1) = \pi/6 \):
\[ \cot(\pi/6) = \log(1) + C \implies \sqrt{3} = 0 + C \implies C = \sqrt{3} \]

Step 4: General solution
\[ \cot(y/x) = \log |x| + \sqrt{3} \]