Consider the initial value problem for y; Laplace?

Consider

y''+8y'+16y=0, y(0)=1, y'(0)=5

find the laplace transform to the solution, that is Y(s)=L(y(t))

Y(s)=_______

Find the function y solution of the IVP abov,

y(t)=__________________

2 Answers

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  • 8 years ago
    Favorite Answer

    (solved in details)

    Consider

    y''+8y'+16y=0, y(0)=1, y'(0)=5

    Take Laplace on both side…we have

    L(y”)+8L(y’)+16L(y)=L(0)

    We have formula(L[y^(k+1)(t)]=s^k+1L[y(t)]-s^ky(0)-s^k-1y'(0)+........)(so according to formula)

    s^2 L [y(t)]-s[y(0)]-y'(0)+8{sL[y(t)]-y(0)}+16L[y(t)]=0

    (taking L[y(t)] common on both side also put the value for y(0)=1 & y'(0)=5 and take to the other side RHS) we have,

    (s^2+8s+16)L[y(t)]=s+8+5=s+13

    L[y(t)]= (s+13)/(s^2+8s+16)=(s+13)/(s+4)^2 ........... (this is your y(s)=Ly(t))

    taking inverse of laplace on both side we have

    y(t)=L^(-1){ [(s+4)+9] / (s+4)^2 }

    y(t)=L^(-1) { 1/(s+4) + 9/(s+4)^2 }

    y(t)= e^(-4t) + 9 e^(-4t) L^(-1)(1/s^2)

    y(t)= e^(-4t) + 9 e^(-4t)* t .......... (this is your final solution )

    (you are in first year B.E )

    Source(s): self
  • ?
    Lv 4
    8 years ago

    I have posted a full solution to your question at one of the math help forums I collaborate with, so that I may give you an easy to read explanation using LaTeX:

    http://www.mathhelpboards.com/f52/ggnores-question...

    Hope this helps

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