Internal problem ID [1210]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN
ORDINARY POINT I. Exercises 7.2. Page 329
Problem number: 6.
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 y^{\prime } x +\frac {y}{4}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 34
Order:=6; dsolve((1+x^2)*diff(y(x),x$2)+2*x*diff(y(x),x)+1/4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}\right ) y \relax (0)+\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 42
AsymptoticDSolveValue[(1+x^2)*y''[x]+2*x*y'[x]+1/4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {147 x^5}{640}-\frac {3 x^3}{8}+x\right )+c_1 \left (\frac {25 x^4}{384}-\frac {x^2}{8}+1\right ) \]