Find the solution of the initial-value problem.y'' + 81y = 0, y(0) = 1, y'(0) = −1

tabita57i

tabita57i

Answered question

2021-05-05

Find the solution of the initial-value problem. y+81y=0,y(0)=1,y(0)=1

Answer & Explanation

Ezra Herbert

Ezra Herbert

Skilled2021-05-06Added 99 answers

Here the initial-value problem is
y+81y=0,y(0)=1,y(0)=1.
The auxiliary equation of the differential equation y+81y=0 becomes
m2+81=0m=9i,9i.
Therefore the general solution is
y(x)=acos9x+bsin9x
Now using the initial condition y(0)=1 we get
a=1⋯(1)
Again using the initial condition y(0)=1 we get
9b=1⋯(2)
From (1) and (2) we have a=1 and b=1/9. Therefore the solution of the initial value problem is
y(x)=cos9x(19)sin9x.

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