Make and solve the given equation x′′ − x′ = 6 + e^{2}t

texelaare 2020-11-08 Answered
Make and solve the given equation xx=6+e2t
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Expert Answer

pattererX
Answered 2020-11-09 Author has 95 answers
Given: xx=6+e2t
Recall: The general solution to a(x)y+b(x)y+c(x)y=g(x) is given by
y=yh+yp,
where y_{h}, the complimentary function is the solution to the homogenens
system a(x)y+b(x)y+c(x)y=0
and yp, the particular solution, is any
function that satisfies the non-homogeneous equation.
Here it is given that yy=6+e2t. Now for the complimentary function,
put y=emx in the homogeneous form
yy=0, which is given by
m2m=0. Therefore
yn=a+bex,
where a, b are arbitrary constant.
Now leeting D=ddx we have
yp=1D2D(6+e2t)
=1D2D6+1D2De2t
=61(D)(1D)1+e2t1222
=61(D)(1+D+D2+)+e2t12
=61(D)+e2t2
=6t+e2t2
Therefore the general solution is given by
y=a+bex+6t+e2t2
where a, b are arbitrary constant.
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Jeffrey Jordon
Answered 2021-10-23 Author has 2087 answers

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