Corresponding homogeneous differential equation
y′′+6y′+9y=16x2 Characteristic (auxiliary) equation
r2+6r+9=0
r1=r2=−3 The general solution of the homogeneous differential equation is
yh=c1e−3x+c2xe−3x Find the particular solution of the non homogeneous differential equation
yp=Ax2+Bx+C
yp′=2Ax+B
yp′′=2A
Substitute
2A+12Ax+6B+9Ax2+9Bx+9C=16x2
A=16/9
B=−64/27
C=32/27
The general solution of the non homogeneous differential equation is
y=c1e−3x+c2xe−3x+916x2−2764x+2732
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