22.6 problem 1(f)

Internal problem ID [5728]

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. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number: 1(f).
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 }-x y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.0 (sec). Leaf size: 29

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

\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{80} x^{6}\right ) y \relax (0)+D\relax (y )\relax (0) x +O\left (x^{8}\right ) \]

Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 34

AsymptoticDSolveValue[(x^2+1)*y''[x]-x*y'[x]+y[x]==0,y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^6}{80}+\frac {x^4}{24}-\frac {x^2}{2}+1\right )+c_2 x \]