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Answer:
The solution of the diferential equation is:
Step-by-step explanation:
Given ; y(0) = 0 ; y'(0) = 0
We need to use the Laplace transform to solve it.
ℒ[y" + y]=ℒ[]
ℒ[y"]+ℒ[y]=ℒ[]
By using the Table of Laplace Transform we get:
ℒ[y"]=s²·ℒ[y]+s·y(0)-y'(0)=s²·Y(s)
ℒ[y]=Y(s)
ℒ[]=
So, the transformation is equal to:
s²·Y(s)+Y(s)=
(s²+1)·Y(s)=
Y(s)=
To be able to separate in terms, we use the partial fraction method:
1=(As+B)(s-1)² + C(s-1)(s²+1)+ D(s²+1)
The equation is reduced to:
1=s³(A+C)+s²(B-2A-C+D)+s(A-2B+C)+(B+D-C)
With the previous equation we can make an equation system of 4 variables.
The system is given by:
A+C=0
B-2A-C+D=0
A-2B+C=0
B+D-C=1
The solution of the system is:
A=1/2 ; B=0 ; C=-1/2 ; D=1/2
Therefore, Y(s) is equal to:
Y(s)=
By using the inverse of the Laplace transform:
ℒ⁻¹[Y(s)]=ℒ⁻¹[]-ℒ⁻¹[
]+ℒ⁻¹[
]