By the elimination of the constants 𝐴 and 𝐵 obtain the differential equation of which 𝑥𝑦 = 𝐴𝑒𝑥 +
𝐵𝑒
−𝑥 + 𝑥
2
is the solution.


To derive the differential equation by eliminating the constants \(A\) and \(B\), we are given the equation:

\[
xy = A e^x + B e^{-x} + x^2
\]

### Step 1: Differentiate the given equation with respect to \(x\)
We'll first differentiate the equation with respect to \(x\) to eliminate the constants \(A\) and \(B\). Apply the product rule for the term \(xy\).

\[
\frac{d}{dx}(xy) = \frac{d}{dx}(A e^x + B e^{-x} + x^2)
\]

On the left-hand side:
\[
\frac{d}{dx}(xy) = y + x \frac{dy}{dx}
\]

On the right-hand side:
\[
\frac{d}{dx}(A e^x + B e^{-x} + x^2) = A e^x - B e^{-x} + 2x
\]

So, after differentiation, we get:
\[
y + x \frac{dy}{dx} = A e^x - B e^{-x} + 2x
\]

### Step 2: Differentiate again to eliminate constants \(A\) and \(B\)
We differentiate the result again to eliminate the constants \(A\) and \(B\).

Differentiate the left-hand side:
\[
\frac{d}{dx}\left(y + x \frac{dy}{dx}\right) = \frac{dy}{dx} + \left(\frac{dy}{dx} + x \frac{d^2y}{dx^2}\right)
\]
Simplifying:
\[
2 \frac{dy}{dx} + x \frac{d^2y}{dx^2}
\]

Differentiate the right-hand side:
\[
\frac{d}{dx}(A e^x - B e^{-x} + 2x) = A e^x + B e^{-x} + 2
\]

Now, equate both sides:
\[
2 \frac{dy}{dx} + x \frac{d^2y}{dx^2} = A e^x + B e^{-x} + 2
\]

### Step 3: Eliminate \(A e^x + B e^{-x}\)
From the original equation \(xy = A e^x + B e^{-x} + x^2\), solve for \(A e^x + B e^{-x}\):

\[
A e^x + B e^{-x} = xy - x^2
\]

Substitute this into the second derivative equation:
\[
2 \frac{dy}{dx} + x \frac{d^2y}{dx^2} = (xy - x^2) + 2
\]

Simplifying:
\[
2 \frac{dy}{dx} + x \frac{d^2y}{dx^2} = xy - x^2 + 2
\]

This is the required differential equation.

### Summary
The differential equation of which \(xy = A e^x + B e^{-x} + x^2\) is the solution is:

\[
2 \frac{dy}{dx} + x \frac{d^2y}{dx^2} = xy - x^2 + 2
\]

Would you like further clarification on any step or details?

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Here are 5 related questions:
1. What is the role of differentiation in eliminating constants from a function?
2. How does the product rule apply in solving differential equations?
3. Why are constants like \(A\) and \(B\) significant in differential equations?
4. How does a second derivative affect the behavior of a differential equation?
5. Can the method used here be applied to other types of equations involving exponentials?

**Tip:** When solving differential equations, always keep track of your constants, as they can often be eliminated through differentiation steps.

By the elimination of the constants A and B obtain the differential equation of which xy = Ae^x + Be^(-x) + x^2 is the solution.

Discover how to eliminate the constants A and B to derive the differential equation from xy = A e^x + B e^(-x) + x^2. This solution involves applying differentiation techniques like the product rule and handling exponential functions. Suitable for undergraduate math students.

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