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Question Number 142570 by mathsuji last updated on 02/Jun/21

$$Findallfunctionsf:\mathbb{R}\to \mathbb{R}suchthat$$$$f(x+y)=2f\left(x\right)+3f\left(y\right)-4xyf(2x-3y)$$$$(\mathrm{\forall }x;y\in \mathbb{R})$$

Answered by ajfour last updated on 02/Jun/21

$$f\left(x\right)=5f\left(\frac{x}{2}\right)-x^{2}f(-\frac{x}{2})$$$$also$$$$f\left(x\right)=2f\left(x\right)+3f\left(0\right)$$$$and$$$$f\left(0\right)=5f\left(0\right)\Rightarrow f\left(0\right)=0$$$$\Rightarrow f\left(x\right)=0$$$$and$$$$f\left(2x\right)=5f\left(x\right)-4x^{2}f(-x)$$$$f(-2x)=5f(-x)-4x^{2}f\left(x\right)$$$$sayf\left(2x\right)=A,f(-2x)=B$$$$f\left(x\right)=\frac{5A+4x^{2}B}{25-16x^4}$$$$f(-x)=\frac{4x^{2}A+5B}{25-16x^4}$$$$\Rightarrow f\left(x\right)=f(-x)$$$$\Rightarrow f\left(2x\right)=(5-4x^2)f\left(x\right)$$$$........$$

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