Solve the following non-homogeneous second order linear differential equation. y''+py'-qy=-rt if p=4, q=8, r=7.

SchachtN 2021-02-09 Answered
Solve the following non-homogeneous second order linear differential equation.
y+pyqy=rt
if p=4, q=8, r=7.
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Expert Answer

tabuordg
Answered 2021-02-10 Author has 99 answers
Substituting p=4, q=8, r=7 in the given equation: y+4y8e=7t. Thus, auxiliary equation of y+4y8y=7t is m2+4m8=0. Solving for m:
m=4±424(8)2
=2±23
Thus the complementary function is yCF=e2t(c1e23t+c2e23t) where c1 and c2 are arbitrary constants.
Now to find Particular integral:
Using the operator D=ddt in y+4y8y=7t
(D2+4D8)y=7t. Thus
yPI=1D2+4D87t
=7811(D2+4D8)(t)
Now using binomial expansion:yPI=7811(D2+4D8)(t)
=78(1+D2+4D8+(D2+4D8)2+...)(t)
=78(t+(D2+4D8)t)
=78(t+12)
=7t8+716
Thus complete solution is yCF+yPI or the complete solution is
e2tc1e23t+c2e23t+7t8+716 where c1 and c2 are arbitrary constant
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