Question

If $f\left(x\right)$ is differentiable function in the interval $(0,\infty )$ such that $f\left(1\right)=1,$ and ${lim}_{t\to x}\frac{t^{2}f\left(x\right)-x^{2}f\left(t\right)}{t-x}=1$ for each $x>0$ then

Answer 1

$\begin{array}{c}{lim}_{t\to x}\frac{t^{2}f\left(x\right)-x^{2}f\left(t\right)}{t-x}\\ {lim}_{t\to x}\frac{2tf\left(x\right)-x^2{f}^{\prime }\left(t\right)}{1}=1\\ 2xf\left(x\right)=x^2{f}^{\prime }\left(x\right)+1\\ 2xy=x^2\frac{dy}{dx}+1\\ \frac{dy}{dx}+\frac{1}{x^2}=\frac{2y}{x}\\ \frac{dy}{dx}+y\left(\frac{-2}{x}\right)=\frac{-1}{x^2}\\ y\times \frac{1}{x^2}=-\int \frac{1}{x^4}dx\\ \frac{y}{x^2}=-\int x^{-4}dx\\ \frac{y}{x^2}=\frac{1x^{-3}}{3}+C\\ 1=\frac{1}{3}+C\\ C=\frac{2}{3}\\ \frac{y}{x^2}=\frac{1}{3x^3}+\frac{2}{3}\end{array}$