we shall solve the partial differential equation
(3D2+10DD′+3D′2)z=ex−y
The auxilliary equation is
3m2+10m+3=0(3m+1)(m+3)=0
3m+1=0 or m+3=0
m=−31,3
C.F=f1(y−31x)+f2(y−3x)
We have the P.I to be
P.I=3D2+10DD′+3D′21ex−y
=3(1)2+10(1)(−1)+3(−1)21ex−y
=3−10+31ex−y
=−41ex−y
z=f1(y−31x)+f2(y−3x)−41ex−y
which is the general solution for the differential equation.
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