If f(x) is differentiable on (0,), f(a)=1, and _t xt^2f(x)-x^2f(t)t-x=1 for each x0, then…

Question

If $f(x)$ is differentiable on $(0,\infty)$, $f(a)=1$, and $\lim_{t	o x}\dfrac{t^2f(x)-x^2f(t)}{t-x}=1$ for each $x>0$, then $f\left(\dfrac32ight)$ is:

Accepted Answer

By L'Hopital: $2xf(x)-x^2f'(x)=1$. Solving this linear ODE gives $f(x)=\dfrac23x^2+\dfrac{1}{3x}$.

$$f\left(\frac32\right) = \frac23\cdot\frac94+\frac13\cdot\frac23 = \frac32+\frac29 = \frac{31}{18}.$$

Solutions · 1