Internal problem ID [5680]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 27
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);
\[ y \relax (x ) = \left (1+x^{2}-\frac {1}{3} x^{4}+\frac {1}{5} x^{6}\right ) y \relax (0)+D\relax (y )\relax (0) x +O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 30
AsymptoticDSolveValue[(1+x^2)*y''[x]+2*x*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^6}{5}-\frac {x^4}{3}+x^2+1\right )+c_2 x \]