Question

When \( y = vx \), the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:

• A. \( \frac{f(v)}{f'(v)} \, dv = \frac{1}{x} \, dx \)
• B. \( f'(v) \, dv = x dx \)
• C. \( \frac{f'(v)}{f(v)} \, dv = \frac{1}{x} \, dx \)
• D. \( f'(v) f(v) \, dv = x dx \)
• E. \( f'(v) f(v) \, dv = -\frac{1}{x} \, dx \)

Hint:

When solving differential equations with substitutions, make sure to correctly apply the chain rule for derivatives, and use the substitution to simplify the equation. This helps in reducing the equation to a more solvable form.

Answer 1

The Correct Option is C

We are given that \( y = vx \), so differentiating both sides with respect to \( x \), we get:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \] Now, substitute \( y = vx \) into the given differential equation: \[ v + x \frac{dv}{dx} = \frac{vx}{x} + \frac{f(v)}{f'(v)}. \] Simplifying: \[ v + x \frac{dv}{dx} = v + \frac{f(v)}{f'(v)}. \] Canceling \( v \) from both sides: \[ x \frac{dv}{dx} = \frac{f(v)}{f'(v)}. \] This can be rewritten as: \[ \frac{f'(v)}{f(v)} \, dv = \frac{1}{x} \, dx. \] Thus, the differential equation reduces to: \[ \frac{f'(v)}{f(v)} \, dv = \frac{1}{x} \, dx. \]
Therefore, the correct answer is option (C).