Key Concepts and Formulas

• Product Rule of Differentiation: $\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)$
• Differential Equations: An equation involving derivatives of a function. Solving it means finding the function.
• Integration: The reverse process of differentiation, used to find the function from its derivative.

Step-by-Step Solution

Step 1: Recognize the Exact Differential

We are given the differential equation: $f(x) + xf'(x) = x^2$

Notice that the left-hand side is the derivative of the product $x f(x)$ with respect to $x$. This is because $\frac{d}{dx}(x f(x)) = f(x) + x f'(x)$ by the product rule. Recognizing this pattern allows us to rewrite the equation.

Step 2: Rewrite the Equation

We rewrite the given equation as: $\frac{d}{dx}(x f(x)) = x^2$

This step is crucial because it transforms a seemingly complex differential equation into a directly integrable form.

Step 3: Integrate Both Sides

Integrate both sides of the equation with respect to $x$: $\int \frac{d}{dx}(x f(x)) \, dx = \int x^2 \, dx$

The integral of the derivative of $x f(x)$ is simply $x f(x)$, and the integral of $x^2$ is $\frac{x^3}{3} + C$, where $C$ is the constant of integration. Thus, we have: $x f(x) = \frac{x^3}{3} + C$

Step 4: Determine the Constant of Integration

We are given that the curve passes through the point $(-2, 2)$. This means that when $x = -2$, $f(x) = 2$. Substitute these values into the equation to find $C$: $(-2)(2) = \frac{(-2)^3}{3} + C$ $-4 = \frac{-8}{3} + C$ $C = -4 + \frac{8}{3} = \frac{-12 + 8}{3} = \frac{-4}{3}$

Step 5: Substitute the Value of C back into the Equation

Substitute $C = -\frac{4}{3}$ back into the equation $x f(x) = \frac{x^3}{3} + C$: $x f(x) = \frac{x^3}{3} - \frac{4}{3}$

Step 6: Rearrange the Equation

Multiply both sides of the equation by 3 to eliminate the fractions: $3x f(x) = x^3 - 4$ Rearrange the terms to get: $x^3 - 3x f(x) - 4 = 0$

This matches option (C) in the original question, however, the provided correct answer is (A). There must be a mistake in the provided correct answer. Let's rework the problem, trying to get to option A. Let's multiply $x f(x) = \frac{x^3}{3} - \frac{4}{3}$ by 3/x. We get: $3f(x) = x^2 - \frac{4}{x}$ $3xf(x) = x^3 - 4$ This still doesn't lead to option A.

Let's assume we made an error in integrating. Let's check if option A satisfies the given equation.

$x^2 + 2xf(x) - 12 = 0$ $2xf(x) = 12 - x^2$ $f(x) = \frac{12 - x^2}{2x}$ $f'(x) = \frac{(-2x)(2x) - (12 - x^2)(2)}{(2x)^2} = \frac{-4x^2 - 24 + 2x^2}{4x^2} = \frac{-2x^2 - 24}{4x^2} = \frac{-x^2 - 12}{2x^2}$

Now, let's plug this into the original differential equation: $f(x) + xf'(x) = x^2$ $\frac{12 - x^2}{2x} + x(\frac{-x^2 - 12}{2x^2}) = x^2$ $\frac{12 - x^2}{2x} + \frac{-x^2 - 12}{2x} = x^2$ $\frac{12 - x^2 - x^2 - 12}{2x} = x^2$ $\frac{-2x^2}{2x} = x^2$ $-x = x^2$ This is not true for all x, so option A is incorrect.

Let's reconsider the integration constant C. $x f(x) = \frac{x^3}{3} + C$ When x=-2, f(x) = 2 $-2 * 2 = \frac{(-2)^3}{3} + C$ $-4 = -\frac{8}{3} + C$ $C = -4 + \frac{8}{3} = \frac{-12 + 8}{3} = -\frac{4}{3}$ Then, $xf(x) = \frac{x^3}{3} - \frac{4}{3}$ $3xf(x) = x^3 - 4$ $x^3 - 3xf(x) - 4 = 0$

The correct answer is option C. There must be a typo in the problem.

Common Mistakes & Tips

• Don't forget the constant of integration! It's a crucial part of solving differential equations.
• Recognizing exact differentials can significantly simplify the problem. Look for patterns like the product rule or chain rule in reverse.
• Double-check your algebra to avoid errors when simplifying and solving for the constant of integration.

Summary

The given differential equation $f(x) + xf'(x) = x^2$ can be rewritten as $\frac{d}{dx}(x f(x)) = x^2$. Integrating both sides gives $x f(x) = \frac{x^3}{3} + C$. Using the given point $(-2, 2)$, we find $C = -\frac{4}{3}$. Substituting this back and rearranging, we get the solution $x^3 - 3xf(x) - 4 = 0$. This corresponds to option (C). The stated correct answer (A) is incorrect.

Final Answer

The final answer is \boxed{x^3 - 3xf(x) - 4 = 0}, which corresponds to option (C).