Limit of an explicitly unsolvable differential equation

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SUMMARY

The discussion centers on solving the initial value problem defined by the differential equation dx/dt = e^((x^2)/2)sin(5x) with the initial condition x(0) = -6. Participants highlight that for x(t) to have a limit L as t approaches infinity, the derivative x'(L) must equal zero, leading to the condition sin(5L) = 0. This indicates that L can be any multiple of π/5, but further analysis is required to determine the stability of these points.

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  • Understanding of differential equations and initial value problems
  • Familiarity with the behavior of trigonometric functions, specifically sine
  • Knowledge of stability analysis in dynamical systems
  • Basic calculus concepts, including limits and derivatives
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Homework Statement


initial value problem:

x(0)=-6

dx/dt=e^((x^2)/2)sin(5x)

find the limit of x(t) as t approaches infinity.

--I'm really unsure as to how to approach this... I was thinking along the lines of seeing where the derivative equalled zero (since if the rate of change is zero it would be indicative of a limit..) but then because of sin(5x) there are so many possibilities to this and I wasn't sure how to incorporate the initial value.

Any help would be really appreciated! Thanks :)
 
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x(t) can only have a limit L if x'(L)=0. That means sin(5L)=0. Which limiting value can you possibly approach? Is it a stable point?
 

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