Question

The solution of the differential equation \[ x\frac{dy}{dx}+y=0 \] passing through the point \((1,1)\) is \(y=\)

• A. \(x^2\)
• B. \(x^{-1}\)
• C. \(x^{-2}\)
• D. \(x\)

Hint:

For equations like \(x\frac{dy}{dx}+y=0\), separate variables and use the given point to find the constant.

Answer 1

The Correct Option is B

Concept: A differential equation can be solved by separating variables if it can be written in the form: \[ \frac{dy}{y}=f(x)\,dx \]

Step 1: Given equation: \[ x\frac{dy}{dx}+y=0 \]

Step 2: Move \(y\) to the other side. \[ x\frac{dy}{dx}=-y \] \[ \frac{dy}{dx}=-\frac{y}{x} \]

Step 3: Separate variables. \[ \frac{dy}{y}=-\frac{dx}{x} \]

Step 4: Integrate both sides. \[ \int \frac{dy}{y}=-\int \frac{dx}{x} \] \[ \log y=-\log x+\log C \]

Step 5: Simplify. \[ \log y+\log x=\log C \] \[ \log(xy)=\log C \] \[ xy=C \] \[ y=\frac{C}{x} \]

Step 6: Use the point \((1,1)\). \[ 1=\frac{C}{1} \] \[ C=1 \] Thus: \[ y=\frac{1}{x}=x^{-1} \] Therefore, \[ \boxed{x^{-1}} \]