Question

The general solution of \[ \frac{dy}{dx} = \frac{1}{3x + 5y} \] is

• A. \( y = Ce^{3x} + \frac{5}{3}x + \frac{1}{3} \)
• B. \( (9x + 15y + 5) = Ke^{3x} \)
• C. \( y = \frac{1}{3} \log(9x + 15y + 5) + C \)
• D. \( x = \frac{5}{3} y + \frac{1}{9} C e^{3y} \)

Hint:

For differential equations of the form \(\frac{dy}{dx} = f(ax + by)\), use substitution \(v = ax + by\) to simplify and separate variables.

Answer 1

The Correct Option is C

Given \[ \frac{dy}{dx} = \frac{1}{3x + 5y} \] This is a homogeneous differential equation. Use the substitution: \[ v = 3x + 5y \] Then, \[ \frac{dv}{dx} = 3 + 5 \frac{dy}{dx} \] Substitute \(\frac{dy}{dx}\) and simplify: \[ \frac{dv}{dx} = 3 + \frac{5}{v} \] Separate variables and integrate: \[ \int v \, dv = \int (3v + 5) \, dx \] Solve this to get the general solution: \[ y = \frac{1}{3} \log(9x + 15y + 5) + C \]