{"type":"rich","version":"1.0","title":"First-order forced equation y'-y=e^x and why the trial solution gains a factor x","html":"<span class=\"wb-embed wb-embed-math\" data-latex=\"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}\">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}</span>","width":640,"height":320}