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: Classifying the differential equation. 
The equation

\[ \frac{dy}{dx} = y \]

is a first-order differential equation where the dependent variable \(y\) and the independent variable \(x\) can be separated. Hence, it belongs to the category of separable differential equations.

Step 2: Separating variables.
Rearrange the equation so that all terms involving \(y\) are on one side and all terms involving \(x\) are on the other:

\[ \frac{1}{y}\,dy = dx \]

Step 3: Integrating both sides.
Integrate with respect to the appropriate variables:

\[ \int \frac{1}{y}\,dy = \int dx \]

This gives:

\[ \ln|y| = x + C \]

where \(C\) is the constant of integration.

Step 4: Expressing the solution explicitly.
Exponentiating both sides to remove the logarithm:

\[ |y| = e^{x+C} \]

The constant \(e^C\) can be replaced by a new arbitrary constant \(C\), giving:

\[ y = Ce^x \]

Step 5: Verifying with the options.
(A) \(y = x + C\): derivative is 1, not equal to \(y\).
(B) \(y = Ce^x\): derivative is \(Ce^x = y\), so it satisfies the equation.
(C) \(y = Cx\): derivative is constant \(C\).
(D) \(y = e^{Cx}\): not the general solution form.

Step 6: Final conclusion.
The general solution of the differential equation is:

\[ \boxed{y = Ce^x} \]