Find the function $f(x)$ so that, for $x\ge 0$, its arc length in the interval $[0, x]$ is $2x+f(x)$.

Question

Find the function $f(x)$ so that, for $x\ge 0$, its arc length in the interval $[0, x]$ is $2x+f(x)$.

I tried to use the formula $$y = \int_0^x \sqrt{1+y'^2} dx$$ with $y = f(x) + 2x$ and $y' = f '(x)+2$ but I'm stuck. Maybe the approach is grouping $x$ and $dx $ on one side of the equation and $y$ and $dy$ on the other side and integrate both sides?

Answer 1

You need to set up the right initial equation: \begin{align*} 2x+f(x)&=\int_0^x\sqrt{1+(f'(\xi))^2}\,d\xi\\ 2+f'(x)&=\sqrt{1+(f'(x))^2}\qquad\text{differentiate both sides}. \end{align*} This can be solved by noting that $f'(x)=C_1$ (you can solve the above equation for $f'(x)$ to find $C_1$) and then integrating to get $f(x)=C_1 x+C_2.$