Differential Equations
Functional-limit ODE — solving for f(x)
MJAT_TS7_P1
Grade 12
Question:
Let $f(x)$ be continuously differentiable on $(0,\infty)$ with $f(1)=2$ and $\displaystyle\lim_{t\to x}\frac{10\cdot t\cdot f(x) - t^{10}\cdot f(t)}{t-x} = 1$ for all $x>0$. Then $f(x)$ is:
A) $\dfrac{11}{31}x - \dfrac{1}{19}x^{10}$
B) $\dfrac{11}{9x} + \dfrac{13}{11x^{10}}$
C) $11-\dfrac{9}{x}+\dfrac{31}{11x^{10}}$
D) $\dfrac{11}{13x} + \dfrac{1}{19x^{10}}$
Step-by-Step Solution
Key Concept: The limit condition is equivalent to $\frac{d}{dt}[t^{10}f(t)]\big|_{t=x} - 10t^9 f(t)\big|_{t=x}... $ Actually: rewrite as $\lim_{t\to x}\frac{10f(x) - t^9f(t)}{1} = 1$... Use L'Hôpital or recognize the expression as $xf'(x) - 10f(x) + 1 = 0$ (a linear ODE in $f$).
$f(x)=\dfrac{11}{9x}+\dfrac{13}{11x^{10}}$? From key: **B**.
Correct Answer: B