Step 1: Understanding the Concept:
This question represents a first-order ordinary differential equation. A differential equation establishes a structural relationship between an unknown mathematical function $y$ and its derivatives. To solve equations where variables can be isolated on opposing sides of the equality symbol, we employ the method of separation of variables.
Step 2: Key Formula or Approach:
The given differential equation is:
$$ \frac{dy}{dx} = y $$
We isolate all terms containing the dependent variable $y$ along with its differential element $dy$ on the left side, and position the independent variable differential element $dx$ on the right side. Once separated, we perform integration on both sides simultaneously:
$$ \int \frac{1}{y} \, dy = \int 1 \, dx $$
The standard integration formula for the reciprocal function is $\int \frac{1}{y} \, dy = \ln|y| + C$.
Step 3: Detailed Explanation:
Let's carry out the separation of variables and subsequent integration actions systematically:
1. Separate the variables: Divide both sides by $y$ and multiply both sides by $dx$:
$$ \frac{1}{y} \, dy = dx $$
2. Integrate both sides:
$$ \int \frac{1}{y} \, dy = \int dx $$
$$ \ln|y| = x + C_1 $$
Where $C_1$ represents the initial arbitrary constant of integration.
3. Solve for $y$ explicitly: Convert the logarithmic expression into its equivalent exponential base form:
$$ |y| = e^{x + C_1} $$
Using the algebraic laws of exponents, we can unpack the right-side addition into a product:
$$ |y| = e^{x} \cdot e^{C_1} $$
4. Simplify the constant term: Since $C_1$ is an arbitrary constant, the term $e^{C_1}$ (or $\pm e^{C_1}$ when removing the absolute value bars) is also just another constant. We can redefine this entire constant factor as $C$:
$$ y = Ce^x $$
This explicit solution corresponds perfectly with option (A).
Step 4: Final Answer:
The correct solution to the differential equation is $y = Ce^x$.