解series solutions for 2nd ODE

╰☆ヾo.海x

Determine the series solutions for the following ODE $$(x^2-1)y''-8xy'+20y=6x^2$$about the regular point $x_0=0$.

icesheep

设 y=a+bx+cx^2+... 代入方程解出系数。

hbghlyj

In[]:= AsymptoticDSolveValue[(x^2-1)y''[x]-8x y'[x]+20y[x]==6x^2,y[x],{x,0,8}]
Out[]= $\displaystyle-\frac{x^4}{2}+\left(1+10 x^2+5 x^4\right) c_1+\left(x+2x^3+\frac{x^5}{5}\right) c_2$


A linear ordinary differential equation can be approximated by a Taylor series expansion $y(x)=\sum_{i} a_{i}\left(x-x_{0}\right)^{i}$ near an ordinary point $x_0$ for the equation. This example shows how to obtain such an approximation using AsymptoticDSolveValue.

hbghlyj

(x^2 - 1) y''[x] - 8 x y'[x] + 20 y[x] == 6 x^2
Differential equation solution $y(x) = c_2 (5 x^4 + 10 x^2 + 1) + c_1 (x - 1)^5 + x^2 + \frac1{10}$
Sample solution family

hbghlyj

The problem states "about the regular point $x_0=0$"
Differential Equations for Engineers (Lebl) 7.2: Series Solutions of Linear Second Order ODEs

Definition: Ordinary and Singular Points
The point \(x_0\) is called an ordinary point if \(p(x_0) \neq 0\) in linear second order homogeneous ODE of the form in Equation 7.2.1. That is, the functions
\[ \dfrac{q(x)}{p(x)} \quad\text{and}\quad \dfrac{r(x)}{p(x)}\]are defined for \(x\) near \(x_0\).
If \( p(x_0)=0\), then we say \(x_0\) is a singular point.

$\frac{-8x}{x^2-1}$ and $\frac{20}{x^2-1}$ are defined for \(x\) near \(0\), so \(0\) is an ordinary point (="regular point")