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Question Number 160659 by ZiYangLee last updated on 04/Dec/21

$$\mathrm{Given}\mathrm{that}{y}^{″}-4y^{\prime }+3y=0\mathrm{and}y\left(0\right)=0,$$$$y^{\prime }\left(0\right)=2,\mathrm{find}y\left(\ln 2\right).$$

Answered by mr W last updated on 04/Dec/21

$$lety={ae}^{px}$$$${ap}^2{e}^{px}-4{ape}^{px}+3{ae}^{px}=0$$$$a(p^2-4p+3)e^{px}=0$$$$p^2-4p+3=0$$$$(p-1)(p-3)=0$$$$\Rightarrow p=1,3$$$$\Rightarrow y={ae}^x+{be}^{3x}$$$$\Rightarrow y^{\prime }={ae}^x+3{be}^{3x}$$$$y\left(0\right)=a+b=0$$$$y^{\prime }\left(0\right)=a+3b=2$$$$\Rightarrow a=-1,b=1$$$$\Rightarrow y=-e^x+e^{3x}$$$$y\left(\ln 2\right)=-e^{\ln 2}+e^{3\ln 2}=-2+8=6$$

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