A differential equation and a nontrivial solution f are given below.

banganX 2021-02-05 Answered

A differential equation and a nontrivial solution f are given below. Find a second linearly independent solution using reduction of order. Assume that all constants of integration are zero.
tx(2t+1)x+2x=0, t>0, f(t)=3e2t

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Expert Answer

lamusesamuset
Answered 2021-02-06 Author has 93 answers

Consider the second linearly independent solution as g(t)=v(t), f(t)=3ve2t
Obtain derivatives:
g(t)=6ve2t+3e2tv
and
g(t)=12ve2t+6e2tv+6e2tv+3e2tv
=12ve2t+12e2tv+3e2tv
Substitute g and its derivative:
t(12ve2t+12e2tv+3e2tv)(2t+1)(6ve2t+3e2tv)+2(3ve2t)=0
12tve2t+12te2tv+3te2tv12tve2t6te2tv6ve2t3e2tv+6ve2t=0
3te2tv+(6t3)e2tv=0
3tv+(6t3)v=0
Further follows,
Let w=v, then
3tw+(6t3)w=0
3tw=(6t3)w
dww=36t3tdt
dww=(1t2)dt
Integrate both sides:
dww=(122)dt
lnw=lnt2t+C
w=elnt2t+C=Ate2t
Then, we have
As w=v, obtain the function v(t):
v=Ate2t
v(t)=A2te2t+A2e2tdt
v(t)=A2te2tA4e2t+C
Choose A=2 and C=0, thus v(t)=te2t+12e2t
Conclusion:
Then, the function g becomes
g(t)=3(te2t+12e2t)e2t
=3t+32
Therefore, the second linearly independent solution is g(t)=3t+32

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