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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/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/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/07%3A_Series_Solutions_of_Linear_Second_Order_Equations/7.06%3A_The_Method_of_Frobenius_IIn this section we begin to study series solutions of a homogeneous linear second order differential equation with a regular singular point at x0=0, so it can be written as x²A(x)y″+xB(x)y′+C(x)y=0, ...In this section we begin to study series solutions of a homogeneous linear second order differential equation with a regular singular point at x0=0, so it can be written as x²A(x)y″+xB(x)y′+C(x)y=0, where A, B, C are polynomials and A(0)≠0.
- 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/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
- https://math.libretexts.org/Courses/Mt._San_Jacinto_College/Differential_Equations_(No_Linear_Algebra_Required)/04%3A__Higher_Order_Linear_ODEs/4.08%3A_Cauchy-Euler_EquationsSubstituting this and Equation ??? into Equation ??? yields Equation ???. Since Equation ??? is the characteristic equation of Equation ???,...Substituting this and Equation ??? into Equation ??? yields Equation ???. Since Equation ??? is the characteristic equation of Equation ???, Theorem 5.2.1 implies that the general solution of Equation ??? on (−∞,∞) is
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/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/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/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 ...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/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/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.