Finding general solution for a nonhomogeneous system of equations { <mtable columnal

tripes3h 2022-07-04 Answered
Finding general solution for a nonhomogeneous system of equations
{ x 1 = x 2 + 2 e t x 2 = x 1 + t 2
I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:
( x 1 x 2 ) = C 1 ( e t + e t e t e t ) + C 2 ( e t e t e t + e t )
Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants C 1 and C 2 a variable.
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Answers (2)

Keegan Barry
Answered 2022-07-05 Author has 18 answers
System is equivalent with
x 2 = x 1 2 e t , x 2 = x 1 + t 2
or
x 2 = x 1 2 e t , x 1 = x 1 + t 2 + 2 e t .
Firstly, by using a method of variations of constants, we will solve the second equation:
x 1 x 1 = t 2 + 2 e t ,
(after that, it will be easy to determine x 2 from the first equation). Solution of homogenuous equation is x 1 = C 1 e t + C 2 e t , so we have to solve the following system:
C 1 e t + C 2 e t = 0 ,
C 1 e t C 2 e t = t 2 + 2 e t ,
where C 1 , C 2 are functions of t. It is easy to get that
C 1 = 1 + t 2 2 e t ,
C 2 ( t ) = C 2 e 2 t 2 e t 2 ( t 2 2 t + 2 ) .
Finally,
x 1 = ( C 1 1 2 ) e t + C 2 e t + t e t t 2 2 ,
x 2 = ( C 1 1 2 ) e t C 2 e t + ( t 1 ) e t 2 t .

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woowheedr
Answered 2022-07-06 Author has 2 answers
If x 1 , x 2 are your solutions to the homogeneous equation (it doesn't actually matter what they are), look for solutions to the inhomogeneous equation in the form y i = v i x i .. If you try that, you will have simple equations for v 1 , v 2 .

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