Question

Solve the differential equation: \[ \frac{dx}{dy} + 2y = e^{3x} \]

• A. \( e^{3x} \)
• B. \( e^{2x} \)
• C. \( e^x \)
• D. \( e^{4x} \)

Hint:

For first-order linear differential equations, after rewriting in standard form, you may identify the form of the solution based on the dominant exponential term. An integrating factor or substitution may be needed to fully solve.

Answer 1

The Correct Option is A

We are given the equation: \[ \frac{dx}{dy} + 2y = e^{3x} \] This is a first-order linear differential equation. To solve this, we can use an integrating factor. Rearrange the equation into standard linear form: \[ \frac{dx}{dy} = e^{3x} - 2y \] This equation suggests that the solution will involve an exponential form related to \( e^{3x} \), and we expect the general solution to be of the form \( e^{3x} \) since this is the main exponential term in the equation. Thus, the correct solution is: \[ \boxed{e^{3x}} \]