Solve dfrac{d^{2}y}{dt^{2}} - 8dfrac{dy}{dt} + 15y=9te^{3t} with y(0)=5, y'(0)=10

nagasenaz 2020-12-06 Answered
Solve dd2ydt2  8ddydt + 15y=9te3t with y(0)=5, y(0)=10
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Expert Answer

toroztatG
Answered 2020-12-07 Author has 98 answers
Given:d2ydt2  8dydt + 15y=9te3t
with y(0)=5, y(0)=10
The auxiliary equation is given by m2  8m + 15=0
solving this we get m=5, 3
Hence the complimentary function is yc=C1e3t + C2e5t
Now The P.I. of given differential equation is
yp=1D2  8D + 159tx3t=1(D  3)(D  5)9te3t
=e3t1(D + 3  3)(D + 3  3)9t
=e3t1D(D  2)9t
=e3t12D(1  D2)9t
= e3t21D(1  D2)1(1  D)19t
= e3t21D(1 + D2 + )9t
= e3t21D(9t + 92)
= e3t21D(92t2+ 92t)
= 94e3t(t + t2)
Hence the general solution is y=yc + yp=C1v3t + C2e5t  92e3t(t + t2)
Now y(0)=0  C1 + e2=5(1)
Also y=3C1e2t + 5C2e5t  94[3e3t(1 + 2t)=3e3t(t + t2)]
Hence y(0)=10  3C1 + 5C2  94=10  12C1 + 20C2=49 (2) solving (1) and (2) we get
C1=518, C2=118
Hence the solution is y=518e3t  118e5t  94e3t(t + t2)
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