First-order forced equation y'-y=e^x and why the trial solution gains a factor x (Q1663): Difference between revisions
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| Property / instance of: mathematical expression / rank | |||
| Property / LaTeX source | |||
| Property / LaTeX source: 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} / rank | |||
| Property / instance of | |||
| Property / instance of: mathematical expression / rank | |||
Normal rank | |||
| Property / LaTeX source | |||
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} | |||
| Property / LaTeX source: 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} / rank | |||
Normal rank | |||
Latest revision as of 07:28, 4 September 2026
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| Language | Label | Description | Also known as |
|---|---|---|---|
| English | 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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