Let f be a twice differentiable non-negative function such that (f(x))^2=25+_0^x((f(t))^2+…

Question

Let $f$ be a twice differentiable non-negative function such that $(f(x))^2=25+\int_0^x\left((f(\mathrm{t}))^2+\left(f^{\prime}(\mathrm{t})\right)^2\right) \mathrm{dt}$

Accepted Answer

Let $y=f(x)$

${y}^{2}=25+{\int }_{x}^{0}(f(t{)}^{2}+{f}^{'}(t{)}^{2})dt$

Differentiating w.r.t. $x$

$\begin{gathered} 2y\frac{dy}{dx}={y}^{2}+{(\frac{dy}{dx})}^{2} \ \Rightarrow {(\frac{dy}{dx}-y)}^{2}=0\Rightarrow \frac{dy}{y}=dx \ \Rightarrow \ln |y|=x+c \end{gathered}$

Such that

$\begin{gathered} f(0{)}^{2}=25+{\int }_{0}^{0}({f}^{2}(t)+{f}^{'}(t{)}^{2})dt \ \Rightarrow f(0)=5 \ \Rightarrow \ln 5=c \end{gathered}$

$\begin{gathered} \Rightarrow \ln |y|=x+\ln 5 \ \Rightarrow \, y=5{e}^{x} \ f(\ln k)=5{e}^{\ln k}=5k \ {5,10,\ldots 625	imes 5} \ =\frac{(5+10+\ldots 5	imes 625)}{625}=\frac{5(625)(626)}{625	imes 2} \ =5	imes 313 \ =1565 \end{gathered}$

Solutions · 1