Key Concepts and Formulas

• Leibniz Integral Rule: $\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} g(x,t) dt \right] = g(x, b(x)) \cdot b'(x) - g(x, a(x)) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} g(x,t) dt$
• Integrating Factor (IF): For $\frac{dy}{dx} + P(x)y = Q(x)$, $IF = e^{\int P(x) dx}$
• General Solution: $y \cdot (IF) = \int Q(x) \cdot (IF) dx + C$

Step-by-Step Solution

Step 1: Differentiating both sides of the equation

We are given the equation $x^2 f(x) - x = 4 \int_0^x t f(t) dt$. We differentiate both sides with respect to $x$ to eliminate the integral.

• Differentiating the left side (LHS): $\frac{d}{dx}(x^2 f(x) - x) = \frac{d}{dx}(x^2 f(x)) - \frac{d}{dx}(x) = 2x f(x) + x^2 f'(x) - 1$ We used the product rule for $x^2 f(x)$.

• Differentiating the right side (RHS): $\frac{d}{dx} \left( 4 \int_0^x t f(t) dt \right) = 4 \frac{d}{dx} \left( \int_0^x t f(t) dt \right) = 4 (x f(x)) = 4x f(x)$ We used the Leibniz integral rule.

• Equating the derivatives: $2x f(x) + x^2 f'(x) - 1 = 4x f(x)$

Step 2: Rearranging into a first-order linear differential equation

We rearrange the equation from Step 1 into the standard form $\frac{dy}{dx} + P(x)y = Q(x)$.

• Rearranging the equation: $x^2 f'(x) - 2x f(x) = 1$

• Dividing by $x^2$ (assuming $x \neq 0$): $f'(x) - \frac{2}{x} f(x) = \frac{1}{x^2}$

• Identifying $P(x)$ and $Q(x)$: Comparing with the standard form, we have $P(x) = -\frac{2}{x}$ and $Q(x) = \frac{1}{x^2}$.

Step 3: Calculating the Integrating Factor (IF)

We calculate the integrating factor using the formula $IF = e^{\int P(x) dx}$.

• Substituting $P(x)$: $IF = e^{\int -\frac{2}{x} dx} = e^{-2 \int \frac{1}{x} dx} = e^{-2 \ln|x|} = e^{\ln(x^{-2})} = x^{-2} = \frac{1}{x^2}$

Step 4: Finding the general solution

We find the general solution using the formula $y \cdot (IF) = \int Q(x) \cdot (IF) dx + C$.

• Substituting $y = f(x)$, $IF = \frac{1}{x^2}$, and $Q(x) = \frac{1}{x^2}$: $f(x) \cdot \frac{1}{x^2} = \int \frac{1}{x^2} \cdot \frac{1}{x^2} dx + C$

• Simplifying the integral: $\frac{f(x)}{x^2} = \int \frac{1}{x^4} dx + C = \int x^{-4} dx + C = -\frac{1}{3x^3} + C$

• Solving for $f(x)$: $f(x) = x^2 \left( -\frac{1}{3x^3} + C \right) = -\frac{1}{3x} + Cx^2$

Step 5: Determining the constant of integration using the initial condition

We use the initial condition $f(1) = \frac{2}{3}$ to find the value of $C$.

• Substituting $x = 1$ and $f(1) = \frac{2}{3}$: $\frac{2}{3} = -\frac{1}{3(1)} + C(1)^2 = -\frac{1}{3} + C$

• Solving for $C$: $C = \frac{2}{3} + \frac{1}{3} = 1$

• The particular solution is: $f(x) = x^2 - \frac{1}{3x}$

Step 6: Calculating the final value $18f(3)$

We calculate $f(3)$ and then find $18f(3)$.

• Substituting $x = 3$: $f(3) = (3)^2 - \frac{1}{3(3)} = 9 - \frac{1}{9} = \frac{81}{9} - \frac{1}{9} = \frac{80}{9}$

• Calculating $18f(3)$: $18f(3) = 18 \cdot \frac{80}{9} = 2 \cdot 80 = 160$

Common Mistakes & Tips

• Remember to use the product rule when differentiating $x^2 f(x)$.
• Pay attention to signs when finding the integrating factor and solving the differential equation.
• Don't forget the constant of integration $C$.

Summary

We started with a functional equation, differentiated it to obtain a first-order linear differential equation, solved the differential equation using the integrating factor method, applied the initial condition to find the particular solution, and finally calculated the value of $18f(3)$.

The final answer is \boxed{160}, which corresponds to option (A).