First-order forced equation y'-y=e^x and why the trial solution gains a factor x (Q1663)

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First-order forced equation y'-y=e^x and why the trial solution gains a factor x
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    Statements

    y'-y=e^{x}:\\qquad y_{c}=C\\,e^{x};\\qquad y_{p}=A\\,x\\,e^{x}\\ \\Rightarrow\\ y_{p}'-y_{p}=A\\,e^{x}=e^{x}\\ \\Rightarrow\\ A=1;\\qquad y=C\\,e^{x}+x\\,e^{x}
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