Question #349865

Undetermined Coefficients

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

Expert's answer

The characteristic (auxiliary) equation for homogeneous differential equation is

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

"r=-1\\pm 2i"

The general solution of the homogeneous differential equation is

Find a particular solution of the nonhomogeneous differential equation.

"y_1'=Ae^x"

"y_1''=Ae^x"

Substitute

"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

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