Question

Let \( f \) be a differentiable function such that \[ 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, \] for \( x \geq 0 \). Then \( f(2) \) is equal to:

Hint:

To solve differential equations involving a product of terms like \( (x + 2) f'(x) \), use separation of variables and then integrate both sides. Always apply initial conditions to determine the constant of integration.

Answer 1

Step 1: Evaluate at $x=0$. Substituting $x=0$ into the given equation yields $2(0 + 2)^2f(0) - 3(0 + 2)^2 = 10 \int_0^0 (t + 2)f(t) dt$. This simplifies to $2(4)f(0) - 3(4) = 0$, then $8f(0) - 12 = 0$, leading to $8f(0) = 12$, and finally $f(0) = \frac{12}{8} = \frac{3}{2}$. Step 2: Differentiate both sides with respect to $x$. Differentiating both sides of the equation with respect to $x$, using the Fundamental Theorem of Calculus, we get $\frac{d}{dx} \left[ 2(x + 2)^2 f(x) - 3(x + 2)^2 \right] = \frac{d}{dx} \left[ 10 \int_0^x (t + 2) f(t) dt \right]$. This expands to $2\left[ 2(x + 2)f(x) + (x + 2)^2 f'(x) \right] - 6(x + 2) = 10(x + 2)f(x)$. Further simplification gives $4(x + 2)f(x) + 2(x + 2)^2 f'(x) - 6(x + 2) = 10(x + 2)f(x)$. Step 3: Simplify the equation. Dividing the equation by $2(x + 2)$ (assuming $x eq -2$) results in $2f(x) + (x + 2)f'(x) - 3 = 5f(x)$. Rearranging yields $(x + 2)f'(x) = 3f(x) + 3$. Step 4: Solve the differential equation. Rearranging the equation gives $\frac{f'(x)}{f(x) + 1} = \frac{3}{x + 2}$. Integrating both sides with respect to $x$ yields $\int \frac{f'(x)}{f(x) + 1} dx = \int \frac{3}{x + 2} dx$. This results in $\ln |f(x) + 1| = 3 \ln |x + 2| + C$, which is equivalent to $\ln |f(x) + 1| = \ln |(x + 2)^3| + C$. Exponentiating both sides gives $|f(x) + 1| = e^C |(x + 2)^3|$. Thus, $f(x) + 1 = K (x + 2)^3$, where $K = \pm e^C$ is a constant. Therefore, $f(x) = K (x + 2)^3 - 1$. Step 5: Find the constant K. Using the known value $f(0) = \frac{3}{2}$, substitute $x = 0$ into the equation: $\frac{3}{2} = K(0 + 2)^3 - 1$. This becomes $\frac{3}{2} = 8K - 1$, then $\frac{5}{2} = 8K$, and finally $K = \frac{5}{16}$. Step 6: Find $f(x)$. Substituting $K$ back into the equation for $f(x)$ gives $f(x) = \frac{5}{16} (x + 2)^3 - 1$. Step 7: Calculate $f(2)$. Substituting $x = 2$ into the equation yields $f(2) = \frac{5}{16} (2 + 2)^3 - 1$. This simplifies to $f(2) = \frac{5}{16} (4)^3 - 1$, then $f(2) = \frac{5}{16} (64) - 1$, leading to $f(2) = 20 - 1$, and finally $f(2) = 19$. Therefore, $f(2) = 19$.