Differential Equations
Linear Differential Equations
JEE Main 2021
Grade 12
Question:
If $\frac{dy}{dx} + y = e^{-x}$ and $y(0) = 0$, then $y$ equals:
(1) $xe^{-x}$
(2) $xe^x$
(3) $e^{-x} + x$
(4) $(x + 1)e^{-x}$
Step-by-Step Solution
Key Concept: This is linear in $y$ with integrating factor $e^{\int 1 \, dx} = e^x$; multiply through by $e^x$ and recognise the left side as $\frac{d}{dx}(ye^x)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)