Question

Consider the differential equation \(\frac{dy}{dx} + \frac{y}{x} = 0\). Choose the CORRECT option(s) for the solution \(y\).

• A. \( y = x + c \); \( c \) is a constant
• B. \( y = \frac{c}{x} \); \( c \) is a constant
• C. \( y = -x + c \); \( c \) is a constant
• D. \( y = 0 \)

Hint:

For separable differential equations, separating variables and integrating will yield the general solution, but don't forget to check for trivial solutions like \( y = 0 \).

Answer 1

The Correct Option is B, D

Step 1: Rearranging the Differential Equation. We start with the equation: \[ \frac{dy}{dx} + \frac{y}{x} = 0 \] Rearranging the terms, we get: \[ \frac{dy}{dx} = -\frac{y}{x} \] This is a separable differential equation. 
Step 2: Separation of Variables and Integration. Separating the variables: \[ \frac{dy}{y} = -\frac{dx}{x} \] Integrating both sides: \[ \int \frac{1}{y} \, dy = -\int \frac{1}{x} \, dx \] This gives: \[ \ln |y| = -\ln |x| + C \] Exponentiating both sides: \[ |y| = \frac{c}{|x|} \] Thus, the solution is: \[ y = \frac{c}{x} \] Step 3: Considering \( y = 0 \). Since \( y = 0 \) satisfies the equation as well, it is also a valid solution.