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Question Number 130893 by EDWIN88 last updated on 30/Jan/21

$$Findaparticularsolution$$$$oftheequationf{}^{\prime }\left(x\right)=f\left(x\right)$$$$suchthatf\left(x\right)=2forx=2$$

Answered by mr W last updated on 30/Jan/21

$$y^{\prime }=y$$$$\frac{dy}{y}=dx$$$$\ln y=x+C$$$$\Rightarrow y={ce}^x$$$$\Rightarrow f\left(2\right)={ce}^2=2\Rightarrow c=2e^{-2}$$$$\Rightarrow f\left(x\right)=y=2e^{x-2}$$

Answered by benjo_mathlover last updated on 30/Jan/21

$${\mathrm{y}}^{\prime }-\mathrm{y}=0\Rightarrow \lambda -1=0;\lambda =1$$$$\mathrm{y}\left(\mathrm{x}\right)=\mathrm{C}{\mathrm{e}}^{\mathrm{x}}\Rightarrow \mathrm{y}\left(2\right)={\mathrm{Ce}}^2=2;\mathrm{C}={2\mathrm{e}}^{-2}$$$$\therefore \mathrm{y}\left(\mathrm{x}\right)={2\mathrm{e}}^{-2}.{\mathrm{e}}^{\mathrm{x}}={2\mathrm{e}}^{\mathrm{x}-2}$$

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