Step 1: Understanding the Question:
We are given a first-order ordinary differential equation and a boundary condition (the solution must pass through \( (1,1) \)).
The equation is of the "variables separable" type. Once we find the general solution containing a constant, we use the point \( (1,1) \) to determine the specific value of that constant.
Step 2: Key Formula or Approach:
1. Separate variables: Put all \( y \) terms on one side and all \( x \) terms on the other.
2. Integrate both sides.
3. Solve for \( y \) and apply boundary conditions.
Step 3: Detailed Explanation:
Step 3.1: Separating variables:
Given \( x \frac{dy}{dx} + y = 0 \).
Rearrange: \( x \frac{dy}{dx} = -y \).
Divide by \( x \cdot y \):
\[ \frac{dy}{y} = -\frac{dx}{x} \]
Step 3.2: Integrating:
\[ \int \frac{dy}{y} = - \int \frac{dx}{x} \]
\[ \ln y = -\ln x + \ln C \]
Where \( \ln C \) is the constant of integration.
Step 3.3: Simplifying the solution:
Using log properties: \( \ln y + \ln x = \ln C \).
\( \ln(xy) = \ln C \).
Taking anti-log: \( xy = C \).
Step 3.4: Finding the constant:
The solution passes through \( (1, 1) \). Substitute \( x = 1 \) and \( y = 1 \):
\[ (1)(1) = C \implies C = 1 \]
Step 3.5: Final Equation:
The specific solution is \( xy = 1 \).
To write it in the form \( y = f(x) \):
\[ y = \frac{1}{x} = x^{-1} \]
Step 4: Final Answer:
The solution of the differential equation is \( y = x^{-1} \).