 Construct a second order DE whose general solution is y(x) =Ax2+BexwhereA, Bare real numbers.  Show your algebra and explain your process.  (Hint:  Compute y′, y′′).  Make sure your final DE is of the forma2(x)y′′+a1(x)y′+a0(x)y= 0 for some functionsa2(x), a1(x), a0(x)

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

Construct a second order DE whose general solution is y(x) = A x^2 + B e^x where A, B are real numbers. Show your algebra and explain your process. (Hint: Compute y′, y′′). Make sure your final DE is of the form a2(x)y′′ + a1(x)y′ + a0(x)y = 0 for some functions a2(x), a1(x), a0(x).

Learn how to construct a second-order differential equation from the general solution y(x) = A x^2 + B e^x, where A and B are real numbers. This step-by-step guide covers derivative computations, algebraic methods, and results in a second-order DE. Suitable for undergraduate students in Calculus II or Differential Equations.

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