Suppose $f$ and $g$ are differentiable functions in $(0,\infty)$ and satisfy $f'(x)(f^5(x) + g^2(x)f(x)) + g'(x) (f^2(x)g(x) + g^3(x))= 0$ show that both f and g are bounded functions
my progress : i realized it can integrated to get that $\frac{f^6(x)}{6} + \frac{g^2(x)f^2(x)}{2} + \frac{g^4(x)}{4} = c$ , where c is some constant , but now how to deciide/check for boundedness ? If the differentiability was given to be in $(-\infty,\infty)$ , then will the answer change ?
In the equation you got, note that each term is nonnegative. Therefore, you have $$|f(x)|\le \sqrt[6]{6c}$$ and $$|g(x)|\le \sqrt[4]{4c}.$$ This proves the boundedness directly.