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The flow invariant for the nonlinear system

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Please solve the following numerical analysis problem:

Determine the flow Qt : R^2 into R^2 for the nonlinear system: x' =f(x) with

f(x) = [ -x1 ]
[ x1^2 + 2x2 ]
and show that the set S = { x E R^2l x2 = -x^2/4 } is invariant with respect to the flow {Qt}.

Please use the step by step method to show the solution.

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https://brainmass.com/math/numerical-analysis/flow-invariant-nonlinear-system-509474

Solution Preview

See the attached file.

Problem: Determine the flow Qt : R^2 into R^2 for the nonlinear system: x' =f(x) with
f(x) = [ -x1 ]
[ x1^2 + 2x2 ]
and show that the set S = { x E R^2l x2 = -x^2/4 } is invariant with respect to the flow {Qt}.

Solution:

Determine Qt:

Start by solving: x1' = -x1 => x1 = Ce^-t.

Plug that result into x2' = x1^2 + 2x2 and solve for x2.
x2' = ...

Solution Summary

This posting provides a solution to a first order linear differential equation initial value problem. It provides a step by step method with the calculations for the flow invariant for a nonlinear system.

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