how to resolve this system of differential equations of order 1? { <mtable columnalign=

wanaopatays

wanaopatays

Answered question

2022-05-22

how to resolve this system of differential equations of order 1?
{ p ˙ 1 = 1 z p 2 p 3 p ˙ 2 = 1 z p 1 p 3 p ˙ 3 = ( 1 y 1 x ) p 1 p 2
where p 1 ( 0 ) = a , p 2 ( 0 ) = b , p 3 ( 0 ) = c and x , y , z are constants.
1) how to resolve this system differential equations by hand?
2) what if x = y = M where M is a constant?

Answer & Explanation

wayembesail

wayembesail

Beginner2022-05-23Added 10 answers

Let's call 1 / z = A and 1 / y 1 / x = B. Multipliying the first equation by p 1 , the second by p 2 and adding we get
p 1 p ˙ 1 + p 2 p ˙ 2 = 0 p 1 2 + p 2 2 = constant = a 2 + b 2 .
Multipliying the first equation by B p 1 , the third by A p 3 and adding we get
B p 1 p ˙ 1 + A p 3 p ˙ 3 = 0 B p 1 2 + A p 3 2 = constant = B a 2 + A c 2 .
We have obtained two first intergrals. We can solve for p 2 and p 3 in terms of p 1 and obtain a first order differential equation for p 1 .

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