Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 155165 by SANOGO last updated on 26/Sep/21

Answered by aleks041103 last updated on 26/Sep/21

$${\int }_0^x(x-t)f^{\prime }\left(t\right)dt={\int }_0^x(x-t)df=$$$$=(x-t)f\left(t\right){\mid }_0^x+{\int }_0^{x}f\left(t\right)dt=xf\left(0\right)+{\int }_0^{x}f\left(t\right)dt$$$$\Rightarrow xf\left(x\right)=x^2+xf\left(0\right)+{\int }_0^{x}f\left(t\right)dt$$$$f\left(x\right)+{xf}^{\prime }\left(x\right)=2x+f\left(0\right)+f\left(x\right)$$$$\Rightarrow f^{\prime }\left(x\right)=2+\frac{f\left(0\right)}{x}$$$$\Rightarrow f\left(x\right)=2x+f\left(0\right)ln\mid x\mid +C$$$$Nowf\left(0\right)=f\left(0\right)ln\mid 0\mid +C...thisisdefined$$$$onlyiff\left(0\right)=0$$$$\Rightarrow f^{\prime }\left(x\right)=2,f\left(0\right)=0$$$$\Rightarrow f\left(x\right)=2x$$$$Check:$$$${\int }_0^x(x-t)f^{\prime }\left(t\right)dt=2{\int }_0^x(x-t)dt=$$$$=2{(xt-\frac{t^2}{2})}_0^x=2x^2-x^2=x^2$$$$\Rightarrow x^2+{\int }_0^x(x-t)f^{\prime }\left(t\right)dt=2x^2=x\left(2x\right)=xf\left(x\right)$$$$OK.$$$$\Rightarrow Ans.f\left(x\right)=2x\Rightarrow f\left(1\right)=2$$

Commented by SANOGO last updated on 26/Sep/21

$$mercibien$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com