The method of frobenius
Download as PPT, PDF7 likes3,940 views
The document discusses the Method of Frobenius for solving ordinary differential equations (ODEs) with singular points. It states that the solution for such an ODE is given as an infinite series involving powers of x. To determine the coefficients in the series, one substitutes the series solution into the original ODE, equates coefficients of like powers of x, and obtains the indical equation. Solving this indical equation gives the indicial solution and recurrence relations for the coefficients.
![∑
∞
=
+
− =−−++−+++−+−
1
10 0}]1)(3)1)((2{[]13)1(2[
n
sn
nn
s
xAAsnsnsnxAsss
1or
0)1)(12(
or
013)1(2
then,0Since
22
1
1
0
−==
=+−
=−+−
≠
ss
ss
sss
A
1,]1)(3)1)((2[ 1 ≥=−++−++ − nAAsnsnsn nn](https://image.slidesharecdn.com/themethodoffrobenius-130719130101-phpapp02/85/The-method-of-frobenius-6-320.jpg)
The method of frobenius
- 1. The Method Of Frobenius
- 2. Methods of Frobenius • If x is not analytic, it is a singular point. )()()()( ''' xfyxqyxpyxr =++ → )( )( )( )( )( )( ''' xr xf y xr xq y xr xp y =++ The points where r(x)=0 are called as singular points.
- 3. Methods of Frobenius (Cont’d) • The solution for such an ODE is given as, ∑ ∞ = = 0m m m r xaxy Substituting in the ODE for values of , )(),(),( ''' xyxyxy equating the coefficient of xm and obtaining the roots gives the indical solution
- 5. 0)(3)1)((2 gets,equationoriginaltheintoSubstitute 0 1 000 =−−++−++ ∑∑∑∑ ∞ = ++ ∞ = + ∞ = + ∞ = + n sn n n sn n n sn n n sn n xAxAxAsnxAsnsn ∑∑ ∞ = + − ∞ = ++ = += 1 1 0 1 Therefore, 1letbyindexofshiftamakeWe n sn n n sn n xAxA nm 0)(3)1)((2 1 1 000 =−−++−++ ∑∑∑∑ ∞ = + − ∞ = + ∞ = + ∞ = + n sn n n sn n n sn n n sn n xAxAxAsnxAsnsn