# ODE resonance clarification

#### fastworld555

[FONT=&quot]I tried using an online calculator for y''-3y'+2y=2x+4e[/FONT][FONT=&quot]3x[/FONT][FONT=&quot] and the answer they gave was y=c1e[/FONT][FONT=&quot]2x[/FONT][FONT=&quot]+c2e[/FONT][FONT=&quot]x[/FONT][FONT=&quot]+x+1.5+2e[/FONT][FONT=&quot]3x[/FONT][FONT=&quot] .However, isn't the yp part wrong as the answer neglected the fact that there is resonance?[/FONT]

#### Archie

There is no resonance. The roots of the characteristic polynomial are 1 and 2, not 3.

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#### fastworld555

I see, but could 2 in 2x not be considered as a root?

#### Archie

No. The right hand side has nothing to do with the characteristic polynomial.

The roots 1 and 2 give terms in $e^{x}$ and $e^{2x}$ to form the complementary solution. These could only resonate with forcing terms (the right hand side) in $e^{x}$ and $e^{2x}$ respectively.

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