Differential Equations
ODE derived from integral condition
MJAT_TS1_P1
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that
$$\int_0^x \bigl(t\cdot f(x) - x\cdot f(t)\bigr)\,t\,dt = 0,\quad \forall\, x\in\mathbb{R}$$
and $f(2) = -\dfrac{1}{3}$. Then:
A) $f(0) = -1$
B) $f(0) = 1$
C) $f(3) = -2$
D) $f(6) = -1$
Step-by-Step Solution
Key Concept: Separate the integral: $f(x)\int_0^x t^2\,dt - x\int_0^x t\,f(t)\,dt = 0$. Differentiating twice or using the structure leads to $f(x)\cdot\frac{x^3}{3} = x\int_0^x t\,f(t)\,dt$. Differentiate to get an ODE: $\frac{f'}{f} = \frac{1}{x}$, so $f(x) = kx$.
$f(x) = kx$ from the ODE. Using $f(2) = 2k = -\frac{1}{3}$ gives $k = -\frac{1}{6}$. So $f(6) = -\frac{6}{6} = -1$. ✓ (D)
Correct Answer: D