Find the sokaton 02 the ger Initial value provsem. y′−y=8t2,y(0)=1

ruigE 2020-11-10 Answered
Find the sokaton 02 the ger Initial value provsem. yy=8t2,y(0)=1
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Expert Answer

faldduE
Answered 2020-11-11 Author has 109 answers

We first solve the homogeneous equation:
yy=0
Its characteristic equation is
r1=0r=1
Therefore, the solution of the homogeneous equation is
yH=Cet
Now we need to find a particular solution. We will try with
yp=(At+B)e2typ,=2(At+B)e2t+Ae2t=(2At+(A+2B))e2t
Thus,
ypyp=8te2t(2At+(A+2B))e2t(At+B)e2t=8te2t(At+(A+B))e2t=8te2t
From this we get
At+(A+B)=8t,
and the system
A=8
A+B=0
Therefore,
A=8B=A=8
and the particular solution is
yp=(8t8)e2t
This means that the total solution is
y=yH+yp=Cet+(8t8)e2t
Now we need to find C. We use the fact that y(0) = 1, so
1=y(0)=Ce0+(808)e20=C8
Therefore, C=9
and the solution of the initial equation is
y=9et+(8t+8)e2t

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