Solving ODEs with Frobenius Method: Troubleshooting and Tips

In summary, the speaker needs help with solving ODEs using the Frobenius method. They encountered difficulties with the first ODE when they had to multiply sine and cosine with y, y', and y''. They also had trouble equating the series in the second ODE due to the presence of cosine. The third ODE needs to be solved using Frobenius, but there is a problem with a1. The speaker tried using y(x)=w(x)/x to find a new solution, and x0=0. The speaker attached the ODE formulas and apologizes for any language barriers, as this is their first time posting on a forum. They request as much help as possible, as they have a deadline for tomorrow
  • #1
edgar
1
0
i need your help...i had to solve some ode using the frobenius method but half-way through i stuck...
for the first ode
i didn't know what to do when i had to multiply the sinx and cosx with y,y',y"
ii)i had a problem because of the cosx and i couldn't equate the series...--->second ode
iii)a) the third is to be solved using frobenius, but a1 has a problem...,
b) by using y(x)=w(x)/x, i had to find the new solution
oh and x0=0
the ode formulas are attached...forgive me i am not a native english speaker and it is the first time i write in this kind of forum...also please help me as much as you can...i have a deadline until tommorow...thanks again...
 

Attachments

  • ode1.bmp
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  • ode2.bmp
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  • ode3.bmp
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Last edited:
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  • #2
Since you are writing y as a Taylor's series, do the same with sine and cosine.
 

1. What is the method of Frobenius for solving problems in ODEs?

The method of Frobenius is a technique used to solve second-order linear differential equations with variable coefficients. It involves assuming a solution of the form y(x) = Σn=0∞ an xn+s, where s is a constant and the coefficients an are to be determined. This method is particularly useful when the differential equation has a singularity at one of the endpoints of the interval of interest.

2. When is the method of Frobenius useful?

The method of Frobenius is useful when the second-order linear differential equation has variable coefficients and a singularity at one of the endpoints of the interval of interest. It is also helpful when the equation cannot be solved using other techniques, such as separation of variables or substitution.

3. How do you determine the value of s in the method of Frobenius?

The constant s in the method of Frobenius is determined by looking at the roots of the indicial equation, which is obtained by substituting y(x) = xs into the differential equation. The value of s will determine the form of the solution, and it is important to choose the correct value in order to obtain the most general solution.

4. What is a recurrence relation in the method of Frobenius?

A recurrence relation is a mathematical equation that defines a sequence of numbers. In the method of Frobenius, the coefficients an in the assumed solution y(x) = Σn=0∞ an xn+s are determined by a recurrence relation. This relation is obtained by substituting the assumed solution into the original differential equation and equating the coefficients of each power of x.

5. Can the method of Frobenius be used to solve higher-order differential equations?

Yes, the method of Frobenius can be extended to solve higher-order linear differential equations with variable coefficients. The assumed solution in this case would be y(x) = Σn=0∞ an xn+s, where s is the constant determined by the roots of the indicial equation. However, the calculations can become more complicated as the order of the differential equation increases, and other techniques may be more efficient in solving higher-order ODEs.

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