Ordinary Differential Equations/Non Homogenous 4

1. $y''-y=e^{-x},x\in \mathbb {R} ,y(0)=y'(0)=0$

Answer: $y(x)={\frac {1}{4}}e^{x}-{\frac {1}{4}}e^{-x}-{\frac {1}{2}}xe^{-x}$

1. $y''-7y+12y=24e^{x}$

Answer: $y_{p}(x)=4e^{-x}$

2. $y''+4y'+4y=e^{-2x}$

Answer: $y_{p}(x)={\frac {1}{2}}x^{2}e^{-2x}$

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