Don't understand x''=x^2-y-xy,\ \ \ y''=y-x as a system of first order

Roger Smith 2022-01-19 Answered
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Expert Answer

Lakisha Archer
Answered 2022-01-19 Author has 39 answers

1. Let, x1=x, x2=x then the x=x2yxy equation converted to x2=x12yx1y where x1=x2.
Let,
y1=y,y2=y then the equation y=yx converted to y2=y1x  where y2=y1.
2. x=tx is already a non-autonomous system. We need to convert it to first order equation.
Let, x1=x, x2=x, x3=x, the above system transformed to x3=tx1 which is a non-autonomous system of first order differential equation.
3. x=tx is not a autonomous system of linear equation. First transfer it to autonomous.
Put, x=tv.
The equation reduces to 3d2vdt2+xvd3vdt3=x2vwheret=xv.
Now let, v=v1, v=v2, v=v3 then the above equation reduces to 3v3+xv1, v3=x2v1 which is an autonomous system of first order differential equation.

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