Step 1: Understanding the Concept:
Solving a differential equation with an initial condition involves two major steps.
First, find the General Solution, which represents a family of curves that all satisfy the differential relationship. This solution will include an arbitrary constant \( C \).
Second, find the Particular Solution by substituting the specific coordinates provided (the boundary condition) into the general solution to solve for the exact numerical value of \( C \).
This specific equation is a "Separable Equation" because the variables \( y \) and \( x \) are easily isolated on opposite sides of the equals sign.
Step 2: Key Formula or Approach:
Separate variables into the form \( \frac{1}{y} dy = \frac{1}{x} dx \).
Integrate both sides using \( \int \frac{1}{u} du = \ln|u| + C \).
Solve for \( C \) using the given point \( (x=1, y=2) \).
Step 3: Detailed Explanation:
Given differential equation:
\[ \frac{dy}{dx} = \frac{y}{x} \]
Separate the variables \( y \) and \( x \):
\[ \frac{1}{y} dy = \frac{1}{x} dx \]
Now, integrate both sides:
\[ \int \frac{1}{y} dy = \int \frac{1}{x} dx \]
Evaluating the integrals:
\[ \ln|y| = \ln|x| + \ln|C| \]
(Note: Using \( \ln|C| \) instead of just \( C \) for the constant of integration makes the simplification cleaner when all other terms are logs).
Applying logarithmic properties (\( \ln A + \ln B = \ln(AB) \)):
\[ \ln|y| = \ln|Cx| \]
Removing the logarithms from both sides:
\[ y = Cx \]
This is our General Solution. Now we use the boundary condition to find the particular solution:
We are given that \( y = 2 \) when \( x = 1 \).
Substitute these values into our general solution:
\[ 2 = C(1) \implies C = 2 \]
Finally, replace the constant \( C \) with its calculated value in the general solution:
\[ y = 2x \]
Step 4: Final Answer:
The particular solution is \( y = 2x \).
This matches Option (A).