Find the general solution of the second order non-homogeneous linear equation: y''-7y'+12y=10sin t+12t+5

emancipezN 2021-02-02 Answered
Find the general solution of the second order non-homogeneous linear equation:
y7y+12y=10sint+12t+5
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Expert Answer

Willie
Answered 2021-02-03 Author has 95 answers

The given second order differential equation is y7y+12y=10sint+12t+5
Find the general solution of the differential equation as follows.
The corresponding homogeneous equation is y7y+12y=0
The auxiliary equation of the corresponding homogeneous equation is
m27m+12=0 Obtain the roots of the auxiliary equation as follows.
m27m+12=0
m24m3m+12=0
m(m4)3(m4)=0
(m4)(m3)=0
m=4, m=3
Therefore, the roots of the auxiliary equation are m1=4 and m2=3
Hence, the complementary solution is yc=c1e4t+c2e3t
Now obtain the particular solution by method of undetermined coefficients as follows.
The choice for the particular solution is yp=Asin(t)+Bcos(t)+Ct+D
This particular solution yp satisfies the given differential equation.
Therefore,
yp7yp+12yp=10sint+12t+5
Asin(t)Bcos(t)7(Acos(t)Bsin(t)+C)
+12(Asin(t)+Bcos(t)+Ct+D)=10sint+12t+5
Asin(t)Bcos(t)7Acos(t)+7Bsin(t)7C+12Asin(t)
+12Bcos(t)+12Ct+12D=10sint+12t+5
(7B+11A)sin(t)+(11B7A)cos(t)+12Ct+12D7C=10sint+12t+5
Equate the coefficients of like terms on both sides and obtain,
7B+11A=10 (1)
11B7A=0 (2)
12C=12 (3)
12D7C=5 (4) Solve the first two equations as shown below
(1)1177B+121A=110
(2)777B+49A=0
170A=110
Then, A=110170=1117
Substitute A=1117 in the equation 7B+11A=10 and solve for B as follows.
7B+11A=10
7B+11(1117)=10
7B+12117=10
7B=17012117
7B=4917
B=717
Now from third equation 12C=12, we have C=1.
Then,
12D7C=5
12D7=5
12D=12
D=1
Hence the particular solution is,
yp=Asin(t)+Bcos(t)+Ct+D

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