To obtain the differential equation whose general solution is given by $xy = a e^x + b e^{-x} + x^2$, where $a$ and $b$ are arbitrary constants, we need to eliminate these constants by differentiating the equation.

Step 1: Differentiate the given equation with respect to $x$

The given equation is: $xy = a e^x + b e^{-x} + x^2$ Differentiate both sides with respect to $x$: $\frac{d}{dx}(xy) = \frac{d}{dx} \left( a e^x + b e^{-x} + x^2 \right)$

Apply the product rule on the left-hand side and differentiate each term on the right-hand side: $y + x \frac{dy}{dx} = a e^x - b e^{-x} + 2x$

Step 2: Differentiate the equation again

Differentiate the equation $y + x \frac{dy}{dx} = a e^x - b e^{-x} + 2x$ again with respect to $x$: $\frac{d}{dx} \left( y + x \frac{dy}{dx} \right) = \frac{d}{dx} \left( a e^x - b e^{-x} + 2x \right)$

Again, apply the product rule on the left-hand side: $\frac{dy}{dx} + \frac{dy}{dx} + x \frac{d^2 y}{dx^2} = a e^x + b e^{-x} + 2$

Simplify the equation: $2 \frac{dy}{dx} + x \frac{d^2 y}{dx^2} = a e^x + b e^{-x} + 2$

Step 3: Eliminate the constants $a$ and $b$

Substitute the expressions for $a e^x + b e^{-x}$ from the differentiated equation: $2 \frac{dy}{dx} + x \frac{d^2 y}{dx^2} = \left( y + x \frac{dy}{dx} - 2x \right) + 2$ $2 \frac{dy}{dx} + x \frac{d^2 y}{dx^2} = y + x \frac{dy}{dx} - 2x + 2$

Simplify this equation: $x \frac{d^2 y}{dx^2} + \frac{dy}{dx} = y + 2$

Step 4: Final Differential Equation

Rearrange the terms to obtain the final form of the differential equation: $x \frac{d^2 y}{dx^2} + \frac{dy}{dx} - y = 2$

This is the differential equation whose general solution is $xy = a e^x + b e^{-x} + x^2$.

Would you like more details or have any questions about the process?

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Here are some related questions you might be interested in:

1. How do you solve a second-order linear differential equation?
2. What methods can be used to find the general solution of a differential equation?
3. How do arbitrary constants affect the solutions of differential equations?
4. Can you explain the method of undetermined coefficients in solving differential equations?
5. What are some common applications of differential equations in physics?

Tip: When differentiating equations with arbitrary constants, it's crucial to differentiate enough times to eliminate all constants and derive the differential equation.