Internal problem ID [6431]
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: 1(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 44
Order:=8; dsolve((1+x^2)*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {5}{24} x^{4}-\frac {17}{144} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{6} x^{5}-\frac {13}{126} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[(1+x^2)*y''[x]+x*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {13 x^7}{126}+\frac {x^5}{6}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {17 x^6}{144}+\frac {5 x^4}{24}-\frac {x^2}{2}+1\right ) \]