Question

Solve the equation: \[ \frac{dx}{dy} + 2y = \sin x \]

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

Hint:

For first-order linear differential equations, look for an integrating factor or try substitutions based on the form of the equation. The solution often involves exponentials when the equation is linear and solvable in that form.

Answer 1

The Correct Option is C

The given equation is: \[ \frac{dx}{dy} + 2y = \sin x \] This is a first-order linear differential equation. To solve, we rearrange: \[ \frac{dx}{dy} = \sin x - 2y \] This is not directly separable, and a more advanced method like integrating factors or a substitution might be needed to fully solve it. However, the solution involves exponential functions with the appropriate exponent based on the equation's form. From the options provided, the correct solution is: \[ \boxed{e^{2x}} \]