The particular integral (yp) of the differential equation (D2 - 6D + 9) y = 7e^-2x is
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Solve (D
2
−6D+9)y=x+e
2x
Medium
Solution
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Given, (D
2
−6D+9)y=x+e
2x
The characteristic equation
p
2
−6p+9=0
⇒(p−3)(p−3)=0
⇒p=3,3 ....(real and equal roots)
C.F. (A+Bx)e
3x
P.I
1
=C
0
+C
1
x
⇒(D
2
−6D+9)(C
0
+C
1
x)=x
⇒−6C
1
+9C
0
+9C
1
x=x
⇒−6C
1
+9C
0
=0
⇒9C
1
=1; 9C
0
=
3
2
⇒C
1
=
9
1
;C
0
=
27
2
⇒P.I.=(
9
x
+
27
2
)
⇒P.I
2
=
D
2
−6D+9
1
e
2x
=
4−−12+9
1
e
2x
=e
2x
The general solution is
y=(A+Bx)e
3x
+
9
x
+
27
2
+e
2x
Step-by-step explanation:
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