In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y′′+6y′+9y=12te−3t−(18t+12) with initial values y(0)=2andy′(0)=−9.

Answer 1

The particular solution of the nonhomogeneous equation is:

yp​(t)=(At+B)te−3t+(Ct+D)e−3t+Et+F

---The first derivative of yp​ is:

yp′​=Ate−3t+Ce−3t+E−3t(At+B)e−3t+(At+B)e−3t−3(Ct+D)e−3t

To solve this second-order linear nonhomogeneous differential equation, we'll first consider the associated homogeneous equation:

y′′+6y′+9y=0

This gives us a repeated root, r = -3.

Therefore, the complementary solution yc​  is of the form:

yc​=(C1​+C2​t)e−3t