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- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.02%3A_Series_Solutions_Near_an_Ordinary_Point_IThis section is devoted to finding power series solutions of (A) in the case where $P_0(0)~\ne~0$. The situation is more complicated if P_0(0)=0; however, if P_1 and P_2 satisfy assumptions that apply...This section is devoted to finding power series solutions of (A) in the case where $P_0(0)~\ne~0$. The situation is more complicated if P_0(0)=0; however, if P_1 and P_2 satisfy assumptions that apply to most equations of interest, then we're able to use a modified series method to obtain solutions of (A).
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.03%3A_Series_Solutions_Near_an_Ordinary_Point_IIIn this section we continue to find series solutions of initial value problems . For the equations considered here it is difficult or impossible to obtain an explicit formula for an in terms of n ....In this section we continue to find series solutions of initial value problems . For the equations considered here it is difficult or impossible to obtain an explicit formula for an in terms of n . Nevertheless, we can calculate as many coefficients as we wish. The next three examples illustrate this.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/12%3A_Series_Solutions_of_Linear_Second_Order_Equations/12.03%3A_Series_Solutions_Near_an_Ordinary_Point_IMany physical applications give rise to second order homogeneous linear differential equations of the form P₀(x)y″+P₁(x)y′+P₂(x)y=0.
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.3%3A_Series_Solutions_and_ConvergenceIn the last section, we saw how to find series solutions to second order linear differential equations. We did not investigate the convergence of these series. In this discussion, we will derive an al...In the last section, we saw how to find series solutions to second order linear differential equations. We did not investigate the convergence of these series. In this discussion, we will derive an alternate method to find series solutions. We will also learn how to determine the radius of convergence of the solutions just by taking a quick glance of the differential equation.
- https://math.libretexts.org/Courses/East_Tennesee_State_University/Book%3A_Differential_Equations_for_Engineers_(Lebl)_Cintron_Copy/7%3A_Power_series_methods/7.2%3A_Series_solutions_of_linear_second_order_ODEsFor linear second order homogeneous ODEs with polynomials as functions can often be solved by expanding functions around ordinary or specific points.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/16%3A_Power_series_methodsThumbnail: The sine function and its Taylor approximations around xo=0 of 5 th and 9 th degree. Contributors and Attributions Jiří Lebl (Oklahoma State University).These pages were supported by N...Thumbnail: The sine function and its Taylor approximations around xo=0 of 5 th and 9 th degree. Contributors and Attributions Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.
- https://math.libretexts.org/Courses/East_Tennesee_State_University/Book%3A_Differential_Equations_for_Engineers_(Lebl)_Cintron_Copy/7%3A_Power_series_methodsThumbnail: The sine function and its Taylor approximations around xo=0 of 5 th and 9 th degree. Contributors Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-090...Thumbnail: The sine function and its Taylor approximations around xo=0 of 5 th and 9 th degree. Contributors Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/03%3A_III._Differential_Equations/09%3A_Series_Solutions/9.01%3A_Ordinary_PointsDeveloping these sequences, we have for the sequence beginning with a0 : \[\begin{aligned} &a_{0} \\ &a_{2}=-\frac{1}{2} a_{0} \\ &a_{4}=-\frac{1}{4 \cdot 3} a_{2}=\frac{1}{4 \cdot 3 \cdot 2} a...Developing these sequences, we have for the sequence beginning with a0 : a0a2=−12a0a4=−14⋅3a2=14⋅3⋅2a0a6=−16⋅5a4=−16!a0 and the general coefficient in this sequence for n=0,1,2,… is a2n=(−1)n(2n)!a0. Also, for the sequence beginning with a1 : \[\begin{aligned} &a_{1} \\ &a_…
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/17%3A_Second-Order_Differential_Equations/17.05%3A_Series_Solutions_of_Differential_EquationsIn some cases, power series representations of functions and their derivatives can be used to find solutions to differential equations.
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.2%3A_Series_Solutions_to_Second_Order_Linear_Differential_EquationsWe have fully investigated solving second order linear differential equations with constant coefficients. Now we will explore how to find solutions to second order linear differential equations whose ...We have fully investigated solving second order linear differential equations with constant coefficients. Now we will explore how to find solutions to second order linear differential equations whose coefficients are not necessarily constant.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.04%3A_Series_Solutions_Near_an_Ordinary_Point_IIIn this section we continue to find series solutions of initial value problems . For the equations considered here it is difficult or impossible to obtain an explicit formula for an in terms of n ....In this section we continue to find series solutions of initial value problems . For the equations considered here it is difficult or impossible to obtain an explicit formula for an in terms of n . Nevertheless, we can calculate as many coefficients as we wish. The next three examples illustrate this.
