Write the following linear differential equations with constant coefficients in the form of the line

Tagiuraoob

Tagiuraoob

Answered question

2022-02-23

Write the following linear differential equations with constant coefficients in the form of the linear system x˙=Ax and solve:
(a)x¨+x˙2x=0
(b)x¨+x=0
(c)dx¨2x¨x˙+2x=0
Hint: Let x1=x,x2=x˙, etc.
I have tried to do this in the following way but I do not know if I am doing well:
Let x1=x,x2=x˙,x3=x¨,x4=x, then the previous system becomes (a)x3+x22x1=0,(b)x3+x1=0,(c)x4x3x2+2x1=0
and thus [x1x2x3x4]=[x23x2x230]. Is this what it should be or is it different? Thank you very much.

Answer & Explanation

Lana Barlow

Lana Barlow

Beginner2022-02-24Added 4 answers

Let x1=x˙,x2=x¨,x3=dx¨. No need to substitute x as we want the final equations to be in the form of Ax. So
(x2+x1=2x),(x2=x),(x32x2x1=2x):
Substituting x2 in the first equation, we get x1=3x. Finally, we substitute x1,x2 in the last equation and we get x3=x. Now solve the final system
(x˙=3x),(x¨=x),(x...=x):

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