We are tasked with constructing a second-order differential equation (DE) whose general solution is given as:

$y(x) = A x^2 + B e^x$

where $A$ and $B$ are real numbers. The final form of the DE should be:

$a_2(x) y'' + a_1(x) y' + a_0(x) y = 0$

Step 1: Compute the First Derivative $y'(x)$

We start by differentiating the given solution $y(x)$ with respect to $x$.

Given: $y(x) = A x^2 + B e^x$

The first derivative is: $y'(x) = \frac{d}{dx}(A x^2 + B e^x) = 2A x + B e^x$

Step 2: Compute the Second Derivative $y''(x)$

Now, we differentiate $y'(x)$ to find the second derivative.

$y''(x) = \frac{d}{dx}(2A x + B e^x) = 2A + B e^x$

Step 3: Construct the DE

We now have the expressions for $y(x)$, $y'(x)$, and $y''(x)$:

$y(x) = A x^2 + B e^x$ $y'(x) = 2A x + B e^x$ $y''(x) = 2A + B e^x$

We aim to form a second