Question

The differential equation is $\frac{dy}{dx} + \frac{y}{x} = 0$ and $y(1) = 2$. Then the value of $y(3) =$

• A. 2
• B. 3
• C. 2/3
• D. 1

Hint:

The equation $y' + y/x = 0$ describes an inverse relationship ($y \propto 1/x$). If $x$ triples, $y$ must become one-third.

Answer 1

The Correct Option is C

Step 1: Concept
Solve the differential equation $\frac{dy}{dx} = -\frac{y}{x}$ using separation of variables.

Step 2: Meaning
$\frac{dy}{y} = -\frac{dx}{x} \implies \log y = -\log x + \log c \implies xy = c$.

Step 3: Analysis
Use the condition $y(1) = 2$: $(1)(2) = c \implies c = 2$. The solution is $xy = 2$ or $y = 2/x$.

Step 4: Conclusion
For $x=3$, $y(3) = 2/3$.
Final Answer: (C)