Question

The general solution of the differential equation $ x \frac{dy}{dx} = y + x \tan\left(\frac{y}{x}\right) $

• A. \( \sin \left( \frac{y}{x} \right) = \frac{C}{x} \)
• B. \( \sin \left( \frac{y}{x} \right) = Cx \)
• C. \( \sin \left( \frac{x}{y} \right) = Cx \)
• D. \( \sin \left( \frac{x}{y} \right) = Cy \)

Hint:

When solving differential equations of the form \( x \frac{dy}{dx} = y + x \tan\left( \frac{y}{x} \right) \), try using substitution methods, such as \( v = \frac{y}{x} \), to simplify and separate the variables.

Answer 1

The Correct Option is B

We are given the differential equation: \[ x \frac{dy}{dx} = y + x \tan\left(\frac{y}{x}\right) \] This is a form of a differential equation that can be solved by the substitution \( v = \frac{y}{x} \). Thus, \( y = vx \), and therefore: \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \] Substitute this into the original equation: \[ x (v + x \frac{dv}{dx}) = vx + x \tan(v) \] Simplify: \[ xv + x^2 \frac{dv}{dx} = vx + x \tan(v) \] Cancel out \( xv \) from both sides: \[ x^2 \frac{dv}{dx} = x \tan(v) \] Now divide both sides by \( x^2 \): \[ \frac{dv}{dx} = \frac{\tan(v)}{x} \] This equation can now be separated: \[ \frac{1}{\tan(v)} dv = \frac{1}{x} dx \] Integrate both sides: \[ \int \frac{1}{\tan(v)} dv = \int \frac{1}{x} dx \] The integral of \( \frac{1}{\tan(v)} \) is \( \ln|\sin(v)| \), and the integral of \( \frac{1}{x} \) is \( \ln|x| \), so we get: \[ \ln|\sin(v)| = \ln|x| + C \] Exponentiate both sides: \[ |\sin(v)| = Cx \] Thus, we have: \[ \sin\left( \frac{y}{x} \right) = Cx \] Therefore, the general solution is \( \sin\left( \frac{y}{x} \right) = Cx \).