Internal problem ID [5727]
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(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+4\right ) y^{\prime \prime }-y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 74
Order:=8; dsolve((4+x^2)*diff(y(x),x$2)-diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{8} x^{2}-\frac {1}{96} x^{3}+\frac {11}{1536} x^{4}+\frac {13}{10240} x^{5}-\frac {533}{737280} x^{6}-\frac {3809}{20643840} x^{7}\right ) y \relax (0)+\left (x +\frac {1}{8} x^{2}-\frac {1}{32} x^{3}-\frac {5}{512} x^{4}+\frac {23}{10240} x^{5}+\frac {283}{245760} x^{6}-\frac {1649}{6881280} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 98
AsymptoticDSolveValue[(4+x^2)*y''[x]-y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {3809 x^7}{20643840}-\frac {533 x^6}{737280}+\frac {13 x^5}{10240}+\frac {11 x^4}{1536}-\frac {x^3}{96}-\frac {x^2}{8}+1\right )+c_2 \left (-\frac {1649 x^7}{6881280}+\frac {283 x^6}{245760}+\frac {23 x^5}{10240}-\frac {5 x^4}{512}-\frac {x^3}{32}+\frac {x^2}{8}+x\right ) \]