Resonance in undetermined coefficients

MediumCalculus~6m

Find a particular solution $y_p$ of $y'' + 3y' + 2y = e^{-x}$ , and report its value at $x=1$ , i.e. $y_p(1)$ . (to 4 decimals)

Show hints (2)+

Check the characteristic roots first: $r^2+3r+2=(r+1)(r+2)$ . The forcing $e^{-x}$ matches $r=-1$ , so the plain guess $Ae^{-x}$ is homogeneous.
Use $y_p=Axe^{-x}$ ; substitute, match the $e^{-x}$ coefficient to get $A=1$ , then evaluate $y_p(1)$ .

Answer

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0.3679 (± 0.001)

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Asked at: Two Sigma