Question

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

• A. \(2\)
• B. \(3\)
• C. \(\frac{2}{3}\)
• D. \(1\)

Hint:

For equations of the form \(\frac{dy}{dx}+\frac{y}{x}=0\), the solution is usually of the form \(xy=C\).

Answer 1

The Correct Option is C

Concept: This is a separable differential equation: \[ \frac{dy}{dx}+\frac{y}{x}=0 \]

Step 1: Given: \[ \frac{dy}{dx}+\frac{y}{x}=0 \] Move the second term to the other side: \[ \frac{dy}{dx}=-\frac{y}{x} \]

Step 2: Separate variables. \[ \frac{dy}{y}=-\frac{dx}{x} \]

Step 3: Integrate both sides. \[ \int \frac{dy}{y}=-\int \frac{dx}{x} \] \[ \log y=-\log x+\log C \]

Step 4: Simplify. \[ \log y+\log x=\log C \] \[ \log(xy)=\log C \] \[ xy=C \] \[ y=\frac{C}{x} \]

Step 5: Use \(y(1)=2\). \[ 2=\frac{C}{1} \] \[ C=2 \] So: \[ y=\frac{2}{x} \]

Step 6: Find \(y(3)\). \[ y(3)=\frac{2}{3} \] Therefore, \[ \boxed{\frac{2}{3}} \]