Find the general solution of the given differential equation.y” + 2y' + y = 2e−t

FizeauV

FizeauV

Answered question

2021-10-18

Find the general solution of the given differential equation.y” + 2y' + y = 2e−t

Answer & Explanation

yunitsiL

yunitsiL

Skilled2021-10-19Added 108 answers

First find solution og homogeneous problem.
r2+2r+1=0r1,2=1
Homogeneous solution:
yc=c1et+c2tet
Let Y=At2et because g(t)=2et from example 5 we know that Y=Aet+Bet is allready described in homogeneous solution.
Plug Y into starting equation to find particular solution:
(At2et)+2(At2et)+(At2et)=2et
(2AtetAt2et)+4Atet2At2et+At2et=2et
2Aet2At2et2Atet+At2et+4Atet2At2et+At2et=2et
2A4At+At2+4At2At2+At2=2
2A=2
A=1
Y=t2et
Solution:
y=yc+Y=c1et+c2tet+t2et
Result: y=c1et+c2tet+t2et

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