Solve non-linear ode system as a function of t. { x ˙ = y y...

kixEffinsoj

kixEffinsoj

Answered

2022-06-25

Solve non-linear ode system as a function of t.
{ x ˙ = y y ˙ = x + x 2 = x ( x 1 )
To solve it as a function x ( y ) or y ( x ) is trivial, but I need the solution as a function of time: ( x ( t ) , y ( t ) ) .
The system has a first integral: H ( x , y ) = 1 2 ( x 2 + y 2 ) 1 3 x 3 . In particular, if no general solution is possible, I'm searching for a solution for H = 1 6 .

Answer & Explanation

luisjoseblash2

luisjoseblash2

Expert

2022-06-26Added 16 answers

Multiplying the 2nd order equation x ¨ = x ( x 1 ) by x ˙ on both sides and integrating with respect to time, we get a nonlinear 1st order equation for x:
x ¨ = x ( x 1 ) x ¨ x ˙ = x ( x 1 ) x ˙ 0 t x ˙ ( τ ) x ¨ ( τ ) d τ = 0 t x ( τ ) ( x ( τ ) 1 ) x ˙ ( τ ) d τ x ˙ 0 x ˙ ( t ) x ˙ d x ˙ = x 0 x ( t ) x ( x 1 ) d x
1 2 ( x ˙ 2 x ˙ 0 2 ) = 1 3 ( x 1 ) 3 + 1 2 ( x 1 ) 2 1 3 ( x 0 1 ) 3 1 2 ( x 0 1 ) 2 x ˙ 2 = x ˙ 0 2 + 2 3 ( x 1 ) 3 + ( x 1 ) 2 2 3 ( x 0 1 ) 3 ( x 0 1 ) 2 = f ( x ; x 0 , x ˙ 0 )
1 2 ( x ˙ 2 x ˙ 0 2 ) = 1 3 ( x 1 ) 3 + 1 2 ( x 1 ) 2 1 3 ( x 0 1 ) 3 1 2 ( x 0 1 ) 2 x ˙ 2 = x ˙ 0 2 + 2 3 ( x 1 ) 3 + ( x 1 ) 2 2 3 ( x 0 1 ) 3 ( x 0 1 ) 2 = f ( x ; x 0 , x ˙ 0 )
Once we have x ˙ 2 = f ( x ; x 0 , x ˙ 0 ), we can at least find an implicit equation for x:
d x d t = f ( x ; x 0 , x ˙ 0 ) x 0 x d χ f ( χ ; x 0 , x ˙ 0 ) = 0 t d τ = t .
The final integral will in general ony be computable in terms of elliptic integrals.

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