Solving ODEs: Is There Any Hope?

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In summary, the conversation discusses the possibility of solving an equation using ordinary generating functions and Picard's method. The analytical solution involves the special function erfi and is expressed on a parametric form. A previous attempt at a formal solution was deleted due to a mistake.
  • #1
James4
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Hello

Comming from Discrete Mathematics, I have very little knowledge in Solving ODEs:

I have the following equation (where E(x) is an ordinary generating function).

[itex]E'(x) = \frac{(E(x)*E(x) +E(x)-x)}{2x*E(x)}[/itex]

with E(0) = 0
Is there any hope to solve this equation?
 
Last edited:
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  • #2
there is always hope. have you looked at picard's method? what do you mean by "solve"? the usual procedure is to give a sequence of approximations that converge to a solution.
 
  • #3
Hello !
The analytical solution involves the special function erfi.
The solution is expressed on a parametric form (see attachment).

The part previously entitled "Formal solution" has been deleted. There was a mistake in it.
 

Attachments

  • Corrected EDO.JPG
    Corrected EDO.JPG
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Related to Solving ODEs: Is There Any Hope?

1. What are ordinary differential equations (ODEs)?

ODEs are mathematical equations that involve one or more independent variables and one or more dependent variables, with the dependent variables being expressed as derivatives of the independent variables. They are commonly used to model physical systems and their behavior over time.

2. Why is solving ODEs challenging?

Solving ODEs can be challenging because they often do not have a closed-form analytical solution and must be solved numerically. Additionally, the complexity of the equations and the boundary conditions can make it difficult to find an accurate solution.

3. What methods are commonly used to solve ODEs?

The most commonly used methods for solving ODEs are numerical methods, such as Euler's method, Runge-Kutta methods, and finite element methods. These methods involve breaking down the ODE into smaller, simpler equations that can be solved iteratively.

4. Is there any hope for finding an exact solution to ODEs?

In some cases, an exact solution to an ODE can be found, but this is often limited to simple equations with specific boundary conditions. For more complex ODEs, it is unlikely that an exact solution can be found and numerical methods must be used.

5. How can I ensure the accuracy of my solution to an ODE?

The accuracy of a solution to an ODE can be improved by using more advanced numerical methods, increasing the number of iterations, and adjusting the initial conditions and boundary conditions. It is also important to double check the solution by comparing it to known solutions or using different methods to solve the same ODE.

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