Question

If \( k \) is a constant, the general solution of \( \dfrac{dy}{dx} - \dfrac{y}{x} = 1\) will be in the form of 
 

• A. \( y = x \ln(kx) \)
• B. \( y = k \ln(kx) \)
• C. \( y = x \ln(x) \)
• D. \( y = x k \ln(k) \)

Hint:

For separable differential equations, separate the variables, integrate, and then solve for the constant using initial conditions if given.

Answer 1

The Correct Option is A

The given differential equation is: \[ \frac{dy}{dx} = \frac{y}{x} \] This is a separable differential equation, and we can solve it by separating the variables. Rearranging the equation, we get: \[ \frac{dy}{y} = \frac{dx}{x} \] Now, integrate both sides: \[ \int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx \] This gives: \[ \ln|y| = \ln|x| + C \] where \( C \) is the constant of integration. Exponentiating both sides: \[ y = e^{\ln|x| + C} = e^C \cdot x \] Since \( e^C \) is just another constant, we can rewrite it as \( k \). Thus, the solution is: \[ y = kx \] However, the solution must also include the factor \( \ln(kx) \), and thus the correct general solution is: \[ y = x \ln(kx) \] Therefore, the correct answer is option (A).
Final Answer: (A) \( y = x \ln(kx) \)