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


The given equation is:

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

We need to eliminate the constants \( A \) and \( B \) to obtain a differential equation for \( y \).

### Step 1: Differentiate with respect to \( x \)

We begin by differentiating both sides of the equation with respect to \( x \):

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

Using the product rule on the left-hand side and differentiating the right-hand side term-by-term:

\[
y + x \frac{dy}{dx} = A e^x - B e^{-x} + 2x
\]

This is our first equation.

### Step 2: Differentiate again

Now, we differentiate this equation once more with respect to \( x \) to eliminate the constants \( A \) and \( B \):

\[
\frac{d}{dx}\left(y + x \frac{dy}{dx}\right) = \frac{d}{dx}\left(A e^x - B e^{-x} + 2x\right)
\]

The left-hand side is:

\[
\frac{dy}{dx} + \frac{d}{dx}\left(x \frac{dy}{dx}\right) = \frac{dy}{dx} + x \frac{d^2y}{dx^2} + \frac{dy}{dx}
\]

So it becomes:

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

The right-hand side is:

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

Thus, the second derivative equation is:

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

### Step 3: Eliminate \( A e^x \) and \( B e^{-x} \)

From the first derivative equation, we already have:

\[
y + x \frac{dy}{dx} = A e^x - B e^{-x} + 2x
\]

To eliminate \( A e^x \) and \( B e^{-x} \), subtract the first equation from the second equation:

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

Simplifying both sides:

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

This simplifies further to:

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

Thus, the required differential equation is:

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

Would you like further clarification on any step or more details?

### 5 Related Questions:
1. What is the general form of a second-order linear differential equation?
2. How can you solve a second-order differential equation like this one?
3. What is the role of boundary conditions in solving differential equations?
4. Can this differential equation be solved using power series methods?
5. How would the solution change if there were no constants \( A \) and \( B \)?

### Tip:
Always check the order and degree of a differential equation before attempting to solve it, as this can guide you towards the appropriate solution method.

By the elimination of the constants A and B, obtain the differential equation of which xy = A e^x + B e^{-x} + x^2 is the solution.

Learn how to derive the differential equation from the equation xy = A e^x + B e^{-x} + x^2 by eliminating the constants A and B. This solution involves differentiation, product rule, and elimination techniques. Ideal for calculus students at the undergraduate level.

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