Question

Solve the differential equation: \( x \frac{dy}{dx} = x \cdot \tan \left( \frac{y}{x} \right) + y \).

Hint:

Use substitution to reduce the given differential equation to a separable form, then integrate both sides.

Answer 1

Step 1: Simplify the equation.
The given equation is: \[ x \frac{dy}{dx} = x \cdot \tan \left( \frac{y}{x} \right) + y \] Divide through by \( x \): \[ \frac{dy}{dx} = \tan \left( \frac{y}{x} \right) + \frac{y}{x} \]

Step 2: Use substitution.
Let \( v = \frac{y}{x} \), so that \( y = vx \). Differentiating both sides with respect to \( x \): \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \] Substitute this into the original equation: \[ v + x \frac{dv}{dx} = \tan(v) + v \] Cancel the \( v \) terms: \[ x \frac{dv}{dx} = \tan(v) \]

Step 3: Solve the resulting equation.
Separate the variables: \[ \frac{dv}{\tan(v)} = \frac{dx}{x} \] Integrating both sides: \[ \int \frac{dv}{\tan(v)} = \int \frac{dx}{x} \] We know that \( \int \frac{dv}{\tan(v)} = \ln|\sin(v)| \) and \( \int \frac{dx}{x} = \ln|x| \), so: \[ \ln|\sin(v)| = \ln|x| + C \] Exponentiating both sides: \[ |\sin(v)| = Cx \] Substitute \( v = \frac{y}{x} \) back into the equation: \[ |\sin \left( \frac{y}{x} \right)| = Cx \] Thus, the solution to the differential equation is: \[ \boxed{|\sin \left( \frac{y}{x} \right)| = Cx} \]