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Sagot :
If the initial value problem is [tex]y^{11} -8y^{1} -15y=0[/tex] and [tex]y^{1}(0) =1[/tex],y(0)=0 then y(t)=[tex](e^{3t} -e^{5t} )/2[/tex].
Given the initial value problem be [tex]y^{11} -8y^{1} -15y=0[/tex]and [tex]y^{1}(0) =1[/tex],y(0)=0.
We are required to find the solution of the given initial value problem.
Laplace transform is an integral transformation that converts a function of a real variable to a function of a complex variable.
Take laplace on the DE, we get
[tex]s^{2}-sY(0)-y^{i}(0)-8[sY(s)-y(0)-15Y(s)]=0[/tex]
[tex]s^{2}Y(s)-s(0)-1-8{sY(s)-0)}+15Y(s)=0[/tex]
(Putting the values given in question)
Y(s)=([tex]s^{2} -8s+15)-1=0[/tex]
Y(s)=1/([tex]s^{2} -8s+15[/tex])
Simplifying the above:
=1/([tex]s^{2} -5s-3s+15)[/tex]
=1/[s(s-5)-3(s-5)]
=1/2 [1/(s-3)-1/(s-5)]
Taking inverse of the above we get,
y(t)=[tex](e^{3t} -e^{5t} )/2[/tex]
Hence if the initial value problem is [tex]y^{11} -8y^{1} -15y=0[/tex] and [tex]y^{1}(0) =1[/tex],y(0)=0 then y(t)=[tex](e^{3t} -e^{5t} )/2[/tex].
Learn more about laplace transform at https://brainly.com/question/17062586
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