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, we get \[ 2(0 + 2)^2f(0) - 3(0 + 2)^2 = 10 \int_0^0 (t + 2)f(t) dt. \] This simplifies to \[ 2(4)f(0) - 3(4) = 0, \] \[ 8f(0) - 12 = 0, \] \[ 8f(0) = 12, \] \[ f(0) = \frac{12}{8} = \frac{3}{2}. \] Step 2: Differentiate both sides with respect to $x$ Differentiating both sides of the equation with respect to $x$, we use the Fundamental Theorem of Calculus: \[ \frac{d}{dx} \left[ 2(x + 2)^2 f(x) - 3(x + 2)^2 \right] = \frac{d}{dx} \left[ 10 \int_0^x (t + 2) f(t) dt \right] \] \[ 2\left[ 2(x + 2)f(x) + (x + 2)^2 f'(x) \right] - 6(x + 2) = 10(x + 2)f(x). \] \[ 4(x + 2)f(x) + 2(x + 2)^2 f'(x) - 6(x + 2) = 10(x + 2)f(x). \] Step 3: Simplify the equation Divide the equation by $2(x + 2)$ (assuming $x \neq -2$): \[ 2f(x) + (x + 2)f'(x) - 3 = 5f(x). \] \[ (x + 2)f'(x) = 3f(x) + 3. \] Step 4: Solve the differential equation Rearrange the equation: \[ \frac{f'(x)}{f(x) + 1} = \frac{3}{x + 2}. \] Integrate both sides with respect to $x$: \[ \int \frac{f'(x)}{f(x) + 1} dx = \int \frac{3}{x + 2} dx. \] \[ \ln |f(x) + 1| = 3 \ln |x + 2| + C. \] \[ \ln |f(x) + 1| = \ln |(x + 2)^3| + C. \] Exponentiate both sides: \[ |f(x) + 1| = e^C |(x + 2)^3|. \] \[ f(x) + 1 = K (x + 2)^3, \] where $K = \pm e^C$ is a constant. \[ f(x) = K (x + 2)^3 - 1. \] Step 5: Find the constant K We know $f(0) = \frac{3}{2}$. Substitute $x = 0$ into the equation: \[ \frac{3}{2} = K(0 + 2)^3 - 1. \] \[ \frac{3}{2} = 8K - 1. \] \[ \frac{5}{2} = 8K. \] \[ K = \frac{5}{16}. \] Step 6: Find f(x) Substitute $K$ back into the equation for $f(x)$: \[ f(x) = \frac{5}{16} (x + 2)^3 - 1. \] Step 7: Calculate f(2) Substitute $x = 2$ into the equation: \[ f(2) = \frac{5}{16} (2 + 2)^3 - 1. \] \[ f(2) = \frac{5}{16} (4)^3 - 1. \] \[ f(2) = \frac{5}{16} (64) - 1. \] \[ f(2) = 20 - 1. \] \[ f(2) = 19. \] Therefore, $f(2) = 19$.