Is this system of differential equations linear? Suppose I have a system of differential equations like below: dotx = x + y + 5, doty = x - y

Kolby Castillo 2022-09-22 Answered
Is this system of differential equations linear?
x ˙ = x + y + 5
y ˙ = x y
Is this system linear or nonlinear differential equations?
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Answers (2)

Nancy Phillips
Answered 2022-09-23 Author has 12 answers
Explanation:
The system is linear, yes. It can be written as
[ x ˙ y ˙ ] = [ 1 1 1 1 ] [ x y ] + [ 5 0 ]
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Medenovgj
Answered 2022-09-24 Author has 1 answers
Step 1
You could write your system of equations as
( x ˙ y ˙ ) = ( 1 1 1 1 ) ( x y ) + ( 5 0 )
and solve it this way:
1. Solve your homogeneous system. In this case, it would be
x h = ( c 1 e t + c 2 e t 2 c 2 e t ) .
2. Solve for your particular solution.
2.1. Rewrite your system as x ( t ) = A x ( t ) + g where g = ( 5 0 ) . Then guess a particular solution. In this case, x p = a .
2.2. Take the derivative,
x p = 0 ,
and plug things in.
0 = A a + g
A a = g .
2.3. Solve the system.
( 1 1 1 1 ) ( a 1 a 2 ) = ( 5 0 )
{ a 1 + a 2 = 5 a 1 a 2 = 0
a 1 = 2.5  and  a 2 = 2.5
a = ( 2.5 2.5 ) .
Step 2
So, the solution is
x ( t ) = ( c 1 e t + c 2 e t 2.5 2 c 2 e t 2.5 ) .
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