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Question Number 144608 by ArielVyny last updated on 26/Jun/21

$$findallaplicationfin\mathbb{R}\to \mathbb{R}f\in C^2$$$$\mathrm{\forall }x\in \mathbb{R}.f^{″}\left(x\right)+f(-x)=x$$

Answered by Olaf_Thorendsen last updated on 27/Jun/21

$$f^{″}\left(x\right)+f(-x)=x\left(1\right)$$$$\mathrm{Let}u\left(x\right)=f\left(x\right)+x$$$$\left(1\right):u^{″}\left(x\right)+u(-x)+x=x$$$$u^{″}\left(x\right)+u(-x)=0$$$$u^{″}(-x)+u\left(x\right)=0$$$$-u^{‴}(-x)+u^{\prime }\left(x\right)=0$$$$u^{\left(4\right)}(-x)+u^{″}\left(x\right)=0$$$$u^{\left(4\right)}(-x)-u(-x)=0$$$$u^{\left(4\right)}\left(x\right)-u\left(x\right)=0$$$$r^4-1=0$$$$r=1,i,-1,-i$$$$u\left(x\right)={\mathrm{C}}_1{e}^x+{\mathrm{C}}_2{e}^{-x}+{\mathrm{C}}_3{e}^{ix}+{\mathrm{C}}_4{e}^{-ix}$$$$\mathrm{or}:$$$$u\left(x\right)=a\cosh x+b\sinh x+c\cos x+d\sin x$$$$$$$$f\left(x\right)=u\left(x\right)-x$$$$f\left(x\right)=a\cosh x+b\sinh x+c\cos x+d\sin x-x$$$$$$$$f^{″}\left(x\right)=a\cosh x+b\sinh x-c\cos x-d\sin x$$$$f(-x)=a\cosh x-b\sinh x+c\cos x-d\sin x+x$$$$\mathrm{With}\left(1\right)\mathrm{necessarily}a=0,d=0$$$$$$$$\mathrm{Finally},\mathrm{the}\mathrm{genaral}\mathrm{solution}\mathrm{is}:$$$$f\left(x\right)=\alpha \sinh x+\beta \cos x-x$$

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