50.18.7 problem 2

Internal problem ID [8101]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 2
Date solved : Monday, January 27, 2025 at 03:43:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 31

Order:=8; 
dsolve((1+x^2)*diff(y(x),x$2)+2*x*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+x^{2}-\frac {1}{3} x^{4}+\frac {1}{5} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 30

AsymptoticDSolveValue[(1+x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]-2*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^6}{5}-\frac {x^4}{3}+x^2+1\right )+c_2 x \]