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- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.04%3A_Regular_Singular_Points_Euler_EquationsThis section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax2y″+bxy′+cy=0, where a, b, and c are constants. This is ...This section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax2y″+bxy′+cy=0, where a, b, and c are constants. This is the simplest equation that satisfies these assumptions.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_(Sloughter)/01%3A_Derivatives/1.10%3A_OptimizationOptimization problems, that is, problems in which we seek to find the greatest or smallest value of some quantity, are common in the applications of mathematics. Because of the extreme-value property,...Optimization problems, that is, problems in which we seek to find the greatest or smallest value of some quantity, are common in the applications of mathematics. Because of the extreme-value property, there is a straightforward algorithm for solving optimization problems involving continuous functions on closed and bounded intervals. Hence we will treat this case first before considering functions on other intervals.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.05%3A_Regular_Singular_Points_Euler_EquationsThis section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax²y′′+bxy′+cy=0, where a, b, and c are constants. This is the simplest equa...This section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax²y′′+bxy′+cy=0, where a, b, and c are constants. This is the simplest equation that satisfies these assumptions.
- 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/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/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/16%3A_Power_series_methods/16.02%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/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/06%3A_Series_Solutions_of_Linear_Second_Order_Equations/6.04%3A_Regular_Singular_Points_Euler_EquationsThis section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax2y″+bxy′+cy=0, where a, b, and c are constants. This is ...This section introduces the appropriate assumptions on P₁ and P₂ in the case where P₀(0)=0, and deals with Euler's equation ax2y″+bxy′+cy=0, where a, b, and c are constants. This is the simplest equation that satisfies these assumptions.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/04%3A_Series_Solutions/4.03%3A_Singular_PointsThe power series method does not always give us the full general solution to a differential equation. Problems can arise when the differential equation has singular points. The simplest equations havi...The power series method does not always give us the full general solution to a differential equation. Problems can arise when the differential equation has singular points. The simplest equations having singular points are Cauchy-Euler equations.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/17%3A_Second-Order_Differential_Equations/17.04%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/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/08%3A_Series_Solutions_of_Linear_Second_Order_Equations/8.04%3A_Series_Solutions_Near_a_Regular_Singular_Point\[\begin{aligned}3x\sum^\infty_{n=0}(n+r)(n+r-1)a_nx^{n+r-2}+\sum^\infty_{n=0}(n+r)a_nx^{n+r-1}-\sum^\infty_{n=0}a_nx^{n+r}=\sum^\infty_{n=0}3(n+r)(n+r-1)a_nx^{n+r-1}+\sum^\infty_{n=0}(n+r)a_nx^{n+r-1...3x∞∑n=0(n+r)(n+r−1)anxn+r−2+∞∑n=0(n+r)anxn+r−1−∞∑n=0anxn+r=∞∑n=03(n+r)(n+r−1)anxn+r−1+∞∑n=0(n+r)anxn+r−1−∞∑n=0anxn+r=0 x∞∑n=0(n+r)(n+r−1)anxn+r−2+∞∑n=0anxn+r=∞∑n=0(n+r)(n+r−1)anxn+r−1+∞∑n=0anxn+r=0
