Question

The solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) is:

• A. $y = cx$
• B. $xy = c$
• C. $y = x+c$
• D. $\log y = x+c$

Hint:

A differential equation where the ratio $y/x$ is constant represents a family of straight lines passing through the origin.

Answer 1

The Correct Option is A

Step 1: Concept
Use the variable separable method.
Step 2: Analysis
$\frac{1}{y} dy = \frac{1}{x} dx$. Integrating both sides: $\int \frac{1}{y} dy = \int \frac{1}{x} dx$.
$\log y = \log x + \log c$.
Step 3: Conclusion
$\log y = \log(cx) \Rightarrow y = cx$.
Final Answer: (A)