12.12.6 problem 6

Internal problem ID [1860]
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
Date solved : Monday, January 27, 2025 at 05:37:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }+\frac {y}{4}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

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);
 

\[ y = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}\right ) y \left (0\right )+\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 42

AsymptoticDSolveValue[(1+x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]+1/4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {147 x^5}{640}-\frac {3 x^3}{8}+x\right )+c_1 \left (\frac {25 x^4}{384}-\frac {x^2}{8}+1\right ) \]