Which of the following is the general solution of $\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0$?

$y = (Ax + B)e^x$

The correct answer is Option (1) → $y = (Ax + B)e^x$ ##

From option (1) we have, $y = (Ax + B)e^x \dots(i)$

On differentiating both sides w.r.t. $x$, we get

$\frac{dy}{dx} = (Ax + B)e^x + e^x(A + 0)$

$\Rightarrow \frac{dy}{dx} = y + Ae^x \dots(ii)$

Again, differentiating both sides w.r.t. $x$, we get

$\frac{d^2y}{dx^2} = \frac{dy}{dx} + Ae^x$

$\Rightarrow \frac{d^2y}{dx^2} = \frac{dy}{dx} + \frac{dy}{dx} - y \quad \left[ \text{from Eq. (ii) } Ae^x = \frac{dy}{dx} - y \right]$

$\Rightarrow \frac{d^2y}{dx^2} = 2 \frac{dy}{dx} - y \Rightarrow \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0$

Hence, option (1) is correct.