Solve this equation: $\frac{f(x)}{\int_{0}^{x_1} f(x)dx}=\frac{g(x)}{\int_{0}^{x_1} g(x)dx}$

Asked Mar 16 '20 at 13:45

Active Mar 16 '20 at 13:50

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Find $f(x)$ and $g(x)$ or some relations between the two functions knowing that $$ \frac{f(x)}{\int_0^{x_1} f(x)dx} = \frac{g(x)}{\int_0^{x_1} g(x)dx}, \quad \forall x \in (0, x_1), \ \forall x_1 \in (0, x_{\max}). $$

One solution is $g(x)=a f(x)$ where $a$ is a constant.

I would be intrested in other solutions less evident.

edited Mar 16 '20 at 13:50

gt6989b

54,422

asked Mar 16 '20 at 13:45

simplex

5

Its useful to write it on the form $\frac{f(x)}{g(x)} = \frac{\int_0^{x_1} f(y)dy}{\int_0^{x_1} g(y)dy}$. Note that the right hand side does not depend on $x$. – Winther Mar 16 '20 at 13:53

Solve this equation: $\frac{f(x)}{\int_{0}^{x_1} f(x)dx}=\frac{g(x)}{\int_{0}^{x_1} g(x)dx}$

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