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Solve-the-ODE-y-2xy-xe-x-2-with-y-0-1-Mastermind-




Question Number 169053 by Mastermind last updated on 23/Apr/22
Solve the ODE   y′ + 2xy = xe^(−x^2 ) , with y(0)=1    Mastermind
$${Solve}\:{the}\:{ODE}\: \\ $$$${y}'\:+\:\mathrm{2}{xy}\:=\:{xe}^{−{x}^{\mathrm{2}} } ,\:{with}\:{y}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by haladu last updated on 23/Apr/22
  y′  + p(x)y = Q(x)     p(x) =  2x          I. f  =  e^(∫2x  dx)   =  e^(x^2  )             y e^(x^2  )   =   ∫  xe^(−x^2 )  ×  e^x^2    dx              ye^x^2     =  ∫  x  dx  =  (x^2 /2)  +  C             y  =  e^(−x^2 )  (  (x^2 /2)  +  C )          x  =  0    y   =  1            1   =   C        ⇒  y  =  e^(−x^2 )  (  (x^2 /2)  + 1  )         Haladu  B    Bello
$$\:\:\boldsymbol{\mathrm{y}}'\:\:+\:\boldsymbol{\mathrm{p}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{y}}\:=\:\boldsymbol{\mathrm{Q}}\left(\boldsymbol{\mathrm{x}}\right) \\ $$$$\:\:\:\boldsymbol{\mathrm{p}}\left(\boldsymbol{\mathrm{x}}\right)\:=\:\:\mathrm{2}\boldsymbol{\mathrm{x}}\:\: \\ $$$$\:\: \\ $$$$\:\:\boldsymbol{\mathrm{I}}.\:\boldsymbol{\mathrm{f}}\:\:=\:\:\boldsymbol{\mathrm{e}}^{\int\mathrm{2}\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{dx}}} \:\:=\:\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:} \:\: \\ $$$$\:\:\: \\ $$$$\:\:\:\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:} \:\:=\:\:\:\int\:\:\boldsymbol{\mathrm{xe}}^{−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:×\:\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\:\boldsymbol{\mathrm{dx}}\:\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\mathrm{ye}}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\:\:=\:\:\int\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{dx}}\:\:=\:\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{2}}\:\:+\:\:\boldsymbol{\mathrm{C}} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:=\:\:\boldsymbol{\mathrm{e}}^{−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\left(\:\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{2}}\:\:+\:\:\boldsymbol{\mathrm{C}}\:\right) \\ $$$$\:\:\: \\ $$$$\:\:\:\boldsymbol{\mathrm{x}}\:\:=\:\:\mathrm{0}\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:=\:\:\mathrm{1}\:\: \\ $$$$\:\:\: \\ $$$$\:\:\:\mathrm{1}\:\:\:=\:\:\:\boldsymbol{\mathrm{C}} \\ $$$$\:\: \\ $$$$\:\:\Rightarrow\:\:\boldsymbol{\mathrm{y}}\:\:=\:\:\boldsymbol{\mathrm{e}}^{−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\left(\:\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{2}}\:\:+\:\mathrm{1}\:\:\right) \\ $$$$\:\:\: \\ $$$$\:\:\boldsymbol{\mathrm{Haladu}}\:\:\boldsymbol{\mathrm{B}}\:\:\:\:\boldsymbol{\mathrm{Bello}} \\ $$

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