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: Understanding the Concept:
This is a variable separable differential equation. We solve it to find the general relationship between \(x\) and \(y\), then use the initial condition to find the constant.
Step 2: Key Formula or Approach:
1. Separate variables: \(\frac{dy}{y} = -\frac{dx}{x}\).
2. Integrate both sides.
3. Solve for \(y\) given \(x=3\).
Step 3: Detailed Explanation:
\[ \frac{dy}{dx} = -\frac{y}{x} \implies \frac{dy}{y} = -\frac{dx}{x} \] Integrating both sides: \[ \ln|y| = -\ln|x| + \ln|C| \] \[ \ln|y| + \ln|x| = \ln|C| \implies xy = C \] Using the condition \(y(1) = 2\): \[ (1)(2) = C \implies C = 2 \] The equation is \( xy = 2 \). To find \(y(3)\), substitute \(x = 3\): \[ 3y = 2 \implies y = \frac{2}{3} \]
Step 4: Final Answer:
The value of \( y(3) \) is \( 2/3 \).