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# Answer to Question #349865 in Differential Equations for Petunia

Question #349865

Undetermined Coefficients

4. y^ prime prime +2y^ prime +5y=1+e^ x

1
2022-06-13T23:06:50-0400
"y''+2y'+5y=1+e^x"

The characteristic (auxiliary) equation for homogeneous differential equation is

"r^2+2r+5=0"

"(r+1)^2=-4"

"r=-1\\pm 2i"

The general solution of the homogeneous differential equation is

"y_h=C_1e^{-x}\\cos 2x+C_2e^{-x}\\sin 2x"

Find a particular solution of the nonhomogeneous differential equation.

"y_1=Ae^x+B"

"y_1'=Ae^x"

"y_1''=Ae^x"

Substitute

"Ae^x+2Ae^x+5Ae^x+5B=1+e^x"

"A=\\dfrac{1}{8}, B=\\dfrac{1}{5}"

"y_1=\\dfrac{1}{8}e^x+\\dfrac{1}{5}"

The general solution of the given differential equation is

"y_h=C_1e^{-x}\\cos 2x+C_2e^{-x}\\sin 2x+\\dfrac{1}{8}e^x+\\dfrac{1}{5}"

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