Limit of an explicitly unsolvable differential equation

In summary, the given problem is an initial value problem where x(0)=-6 and dx/dt=e^((x^2)/2)sin(5x). The task is to find the limit of x(t) as t approaches infinity. It is suggested to find the value of x where the derivative x'(L) is equal to 0, which in this case is when sin(5L)=0. This means that x(t) can only have a limit if it approaches a stable point.
  • #1
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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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  • #2
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?
 

1. What is a limit of an explicitly unsolvable differential equation?

A limit of an explicitly unsolvable differential equation refers to the value that the dependent variable approaches as the independent variable approaches a certain value. It is often represented using mathematical notation as lim f(x), where f(x) is the function representing the differential equation.

2. How is the limit of an explicitly unsolvable differential equation different from a regular limit?

The limit of an explicitly unsolvable differential equation differs from a regular limit in that it involves a function that cannot be solved for explicitly. This means that the function cannot be expressed in terms of elementary functions such as polynomials, trigonometric functions, or exponential functions.

3. Can the limit of an explicitly unsolvable differential equation be evaluated numerically?

Yes, the limit of an explicitly unsolvable differential equation can be evaluated numerically using methods such as Euler's method or the Runge-Kutta method. These methods use a series of approximations to find the value of the limit.

4. What are some real-world applications of explicitly unsolvable differential equations?

Explicitly unsolvable differential equations are commonly used in physics, engineering, and other scientific fields to model complex systems and phenomena. Some examples include the motion of planets in the solar system, the spread of diseases in a population, and chemical reactions.

5. Is it possible to find an exact solution to an explicitly unsolvable differential equation?

No, it is not possible to find an exact solution to an explicitly unsolvable differential equation. However, numerical methods can be used to find approximate solutions that are accurate to a certain degree. Additionally, in some cases, it is possible to find an analytical solution to a simplified version of the differential equation.

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