Method of Frobenius with exponential coefficients

In summary, the method of Frobenius can be used on variable coefficient ODE's when the coefficients are powers of the independent variable. For the equation y'' + e-xy = 0, the solution would be in the form of y = ∑anxn, where the exponential term can be expanded into a series and the Cauchy product can be used. However, a specific solution for this equation is already known.
  • #1
phil ess
70
0
I know how to do Frobenius on variable coefficient ODE's but only when the coefficients are powers of the independent variable. Can I do method of Frobenius on something like:

y'' + e-xy = 0 ?

What form would I assume a solution of? Just the regular y=sum(Akxk+r ?

Thanks for the help!
 
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  • #2
I think you would let y = ∑anxn, expand the exponential in a series about 0 and use the Cauchy product for the e-xy term. You should be able to calculate the first few terms by hand and with luck maybe find a pattern.

If it is that specific equation you are interested in, the solution is known. See

http://eqworld.ipmnet.ru/en/solutions/ode/ode0227.pdf
 

FAQ: Method of Frobenius with exponential coefficients

1. What is the Method of Frobenius with exponential coefficients?

The Method of Frobenius with exponential coefficients is a mathematical technique used to solve certain types of differential equations. It involves expressing the solution as a series of exponential functions and finding coefficients that satisfy the given equation.

2. When is the Method of Frobenius with exponential coefficients used?

This method is typically used to solve linear differential equations with variable coefficients, particularly when the coefficients involve exponential functions.

3. How does the Method of Frobenius with exponential coefficients work?

The method involves substituting a series solution of the form y = ∑n=0∞ an xn into the given differential equation. The coefficients an are then determined by solving a system of equations resulting from the substitution.

4. What are the benefits of using the Method of Frobenius with exponential coefficients?

This method allows for the solution of differential equations that cannot be solved by other means. It also provides a general solution that can be used to find particular solutions for specific initial conditions.

5. Are there any limitations to the Method of Frobenius with exponential coefficients?

Yes, this method can only be used for certain types of differential equations, and it may not always yield a solution, particularly if the coefficients involve complex numbers. It also requires a certain level of mathematical knowledge and skill to apply effectively.

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