Question

If \[ \frac{dy}{dx} = y \] then which of the following is the correct solution?

• A. \( y = Ce^x \)
• B. \( y = Cx \)
• C. \( y = x^2 + C \)
• D. \( y = \log x \)

Hint:

Any process where the growth rate of a quantity is proportional to its size yields an exponential function.
Thus, the equation \( \frac{dy}{dx} = ky \) always has the standard solution \( y = Ce^{kx} \).
Remembering this standard model saves valuable time during competitive exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The given equation is a first-order separable differential equation where the rate of change of \( y \) is directly proportional to \( y \) itself.
We need to find the function \( y(x) \) that satisfies this relationship.

Step 2: Key Formula or Approach:
We use the method of separation of variables.
We rearrange the differential equation so that all terms containing \( y \) are on the left-hand side, and all terms containing \( x \) are on the right-hand side:
\[ \frac{1}{y} \, dy = dx \]
Then, we integrate both sides using standard integration formulas:
\[ \int \frac{1}{y} \, dy = \int dx \]
We use the logarithmic integration rule: \( \int \frac{1}{y} \, dy = \ln|y| \).

Step 3: Detailed Explanation:
Starting with the separated differential equation:
\[ \frac{dy}{y} = dx \]
Integrating both sides gives:
\[ \int \frac{1}{y} \, dy = \int 1 \, dx \]
Evaluating both integrals yields:
\[ \ln|y| = x + C_1 \]
where \( C_1 \) is the constant of integration.
To solve for \( y \), we exponentiate both sides (using base \( e \)):
\[ e^{\ln|y|} = e^{x + C_1} \] \[ |y| = e^x \cdot e^{C_1} \]
Since \( e^{C_1} \) is a constant, we can define a new arbitrary constant \( C = \pm e^{C_1} \).
This simplifies the expression to:
\[ y = Ce^x \]
Let us verify this solution by differentiating it:
\[ \frac{dy}{dx} = \frac{d}{dx}(Ce^x) = Ce^x = y \]
This confirms that the differential equation is satisfied.
Let us analyze why other options are incorrect:
- Option (B) \( y = Cx \) gives \( \frac{dy}{dx} = C \), which is not equal to \( y \).
- Option (C) \( y = x^2 + C \) gives \( \frac{dy}{dx} = 2x \), which is not equal to \( y \).
- Option (D) \( y = \log x \) gives \( \frac{dy}{dx} = \frac{1}{x} \), which is not equal to \( y \).

Step 4: Final Answer:
The general solution of the differential equation is \( y = Ce^x \), which corresponds to Option (A).