Solve left(d^{2}frac{y}{dt^{2}}right) + 7left(frac{dy}{dt}right) + 10y=4te^{-3}t with y(0)=0, y'(0)= -1

Braxton Pugh 2021-02-15 Answered
Solve
(d2ydt2) + 7(dydt) + 10y=4te3t with
y(0)=0, y(0)= 1
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Expert Answer

Benedict
Answered 2021-02-16 Author has 108 answers

Given:
d2ydt2 + 7dydt + 10y=4te3t
with y(0)=0, y(0)= 1
The auxiliary equation is given by m2 + 7m + 10=0
solving this we get m= 5, 2
Hence the complimentary function is yc=C1e2t + C2e5t
Now The P.I. of given differential equation is
yp=1D2 + 7D + 104tx3t=1(d + 5)(D + 2)4te3t
=e3t1(D  3 + 5)(D  3 + 2)4t
=e3t1(D + 2)(D  1)4t
=e3t12(1 + D2)(1  D)4t
= e3t2(1 + D2)1(1  D)14t
= e3t2(1  D6 + )(1 + D + D2)4t
= e3t2(1 + (1  12)D + )4t
= e3t2(4t + 2)
= e3t(2t + 1)
Hence the general solution is y=yc+yp=C1v2t+C2e5te3t(2t+1)
Now y(0)=0  C1 + e2  1=0  C1=C2=1 (1)
Also y= 2C1e2t  5C2e5t + 3e3t(2t + 1)  2e3t
Hence y(0)= 1 2C1  5C2  2 + 3= 1  2C1 + 5C2=2  (2)
solving (1) and (2) we get C1=1, C2=0
Hence the solution is y=e2t  e3t(2t + 1)

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