Given the simultaneous differential equations,
(D+2)x−3y−3x+(D+2)y=1(1)=e−t (2)
Applying (D+2) operator for (1), Multiplying (2) by 3 and adding we get,
(D+2)2x−3(D+2)y−9x+3(D+2)y(D+2)2x−9x(D2+4D+4−9)x(D2+4D−5)x=2(1)=3e−t (2)=2+3e−t=2+3e−t=2+3e−t
The auxiliary equation is, m2+4m−5=0. Solving we get m=−5,1.
The complementary function for x(t) is, C.F=c1e−5t+c2et
Particular Integral for x(t)=D2+4D−51(2+3e−t)=−52−83e−t(Replacing D by 0 and -1 for first and second term respectively)
∴x(t)=c1e−5t+c2et−(52+83e−t)
From equation (1), 3y=Dx+2x−1
3y3y3y∴y(t)=D(c1e−5t+c2et−(52+83e−t))+2(c1e−5t+c2et−(52+83e−t))−1=−5c1e−5t+c2et+83e−t+2c1e−5t+2c2et−54+43e−t−1=−3c1e−5t+3c2et−83e−t−59=−c1e−5t+c2et−8e−t−53
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