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Q.
A function $f(x)$ satisfying $\int\limits_0^1 f(t x) d t=n f(x)$, where $x>0$, is -
Differential Equations
Solution:
$\int\limits_0^1 f(t x) d t=n f(x)$
Let $ t x=u \Rightarrow d t=\frac{d u}{x}$
$\therefore \frac{1}{x} \int\limits_0^x f(u) d u \Rightarrow n f(x) \Rightarrow \int\limits_0^x f(u) d u=n x f(x)$
$f(x)=n\left[f(x)+x f^{\prime}(x)\right] \Rightarrow f(x)\left(\frac{1-n}{n}\right)=x f^{\prime}(x)$
$\int \frac{d x}{x}=\frac{n}{1-n} \int \frac{d y}{y} \Rightarrow \frac{1-n}{n} \ell n x=\ell n y+\ell n c$
$x^{\frac{1-n}{n}}=c y \Rightarrow y=c^{\prime} x^{\frac{1-n}{n}}$