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.