Question

Find the complementary function and partial derivative for this equation (D^2+2D+5)y=xe^x.​

Answer 1

Answer: e^(-x)(Acos(2x)+Bsin(2x))

Step-by-step explanation:

Complementary function the solution of the corresponding Homogeneous equation, i.e, (D^2+2D+5)y=0------(1)

Now you write auxiliary equation for (i), which is m^2+2m+5=0 (replace the operator sign D with a variable)  

Its roots are (-1+2i) and (-1-2i). Hence the Complementary function is y=e^(-x)(Acos(2x)+Bsin(2x)), where A and B are arbitrary constants. (Note that if a+bi and a-bi are the roots of the auxiliary equation then the corresponding complemetary function is e^(ax)(Acos(bx)+Bsin(bx))).