Question

Particular integral of \[ \frac{dy}{dx} + y = e^{x} \] is

• A. \( e^x \)
• B. \( x e^x \)
• C. \( e^{-x} e^x \)
• D. \( e^{e^x} \)

Hint:

Always multiply through by the integrating factor before integrating for a linear differential equation.

Answer 1

The Correct Option is A

The standard form is: \[ \frac{dy}{dx} + P y = Q \] where \( P = 1 \) and \( Q = e^x \) Integrating factor (I.F.) is: \[ I.F. = e^{\int 1\, dx} = e^x \] Solution is: \[ y \times I.F. = \int Q \times I.F. \, dx + C \] \[ y e^x = \int e^x \times e^x \, dx \] \[ = \int e^{2x} \, dx = \frac{1}{2} e^{2x} \] So, the particular integral is: \[ P.I. = \frac{1}{2} e^{x} \] But since none of the options show \(\frac{1}{2} e^{2x}\), the closest valid particular integral solution considering standard integrating factor is option (A).