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y-3y-2y-0-y-0-y-3-for-x-0-




Question Number 94969 by bobhans last updated on 22/May/20
y′′−3y′+2y=0 ; y=0,y′=3 for x=0
$$\mathrm{y}''−\mathrm{3y}'+\mathrm{2y}=\mathrm{0}\:;\:\mathrm{y}=\mathrm{0},\mathrm{y}'=\mathrm{3}\:\mathrm{for}\:\mathrm{x}=\mathrm{0} \\ $$
Answered by bobhans last updated on 22/May/20
homogenous solution  υ^2 −3υ+2 = 0 ⇒(υ−2)(υ−1)=0   { ((υ=2)),((υ=1)) :} ⇒y_h  = Ae^(2x) +Be^x     { ((0=A+B ⇒A=−B)),((y′=2Ae^(2x) +Be^x  ⇒3=2A+B)) :}  3 = −2B+B  { ((B=−3)),((A=3)) :}  y = 3e^(2x) −3e^x
$$\mathrm{homogenous}\:\mathrm{solution} \\ $$$$\upsilon^{\mathrm{2}} −\mathrm{3}\upsilon+\mathrm{2}\:=\:\mathrm{0}\:\Rightarrow\left(\upsilon−\mathrm{2}\right)\left(\upsilon−\mathrm{1}\right)=\mathrm{0} \\ $$$$\begin{cases}{\upsilon=\mathrm{2}}\\{\upsilon=\mathrm{1}}\end{cases}\:\Rightarrow\mathrm{y}_{\mathrm{h}} \:=\:\mathrm{Ae}^{\mathrm{2x}} +\mathrm{Be}^{\mathrm{x}} \: \\ $$$$\begin{cases}{\mathrm{0}=\mathrm{A}+\mathrm{B}\:\Rightarrow\mathrm{A}=−\mathrm{B}}\\{\mathrm{y}'=\mathrm{2Ae}^{\mathrm{2x}} +\mathrm{Be}^{\mathrm{x}} \:\Rightarrow\mathrm{3}=\mathrm{2A}+\mathrm{B}}\end{cases} \\ $$$$\mathrm{3}\:=\:−\mathrm{2B}+\mathrm{B}\:\begin{cases}{\mathrm{B}=−\mathrm{3}}\\{\mathrm{A}=\mathrm{3}}\end{cases} \\ $$$$\mathrm{y}\:=\:\mathrm{3e}^{\mathrm{2x}} −\mathrm{3e}^{\mathrm{x}} \: \\ $$

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