Question

The general solution of $\displaystyle \frac{dy}{dx} = y$ is:

• A. $y = x + C$
• B. $y = Ce^{x}$
• C. $y = Cx$
• D. $y = e^{Cx}$

Hint:

For equations of the form $\frac{dy}{dx} = ky$, the general solution is always $y = Ce^{kx}$.

Answer 1

The Correct Option is B

Step 1: Identifying the type of differential equation.
The given equation \[ \frac{dy}{dx} = y \] is a first-order differential equation in which variables can be separated. Hence, it is a separable differential equation.
Step 2: Separating the variables.
Rewriting the equation, we get: \[ \frac{1}{y}\,dy = dx \] This separates all $y$ terms on one side and all $x$ terms on the other side.
Step 3: Integrating both sides.
Integrating both sides, \[ \int \frac{1}{y}\,dy = \int dx \] which gives \[ \ln |y| = x + C \] where $C$ is the constant of integration.
Step 4: Removing the logarithm.
Taking exponential on both sides, \[ |y| = e^{x+C} \] This can be written as \[ y = Ce^{x} \] where $C$ is an arbitrary constant (positive or negative).
Step 5: Analysis of the given options.
(A) $y = x + C$: Incorrect — this satisfies $\frac{dy}{dx} = 1$, not $y$.
(B) $y = Ce^{x$:} Correct — differentiating gives $\frac{dy}{dx} = Ce^{x} = y$.
(C) $y = Cx$: Incorrect — derivative is constant $C$.
(D) $y = e^{Cx$:} Incorrect — this does not represent the general solution form.
Step 6: Conclusion.
The general solution of the differential equation $\displaystyle \frac{dy}{dx} = y$ is \[ \boxed{y = Ce^{x}} \]