12.12.25 problem 27

Internal problem ID [1879]
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 : 27
Date solved : Monday, January 27, 2025 at 05:37:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 35

Order:=6; 
dsolve((1+x^2)*diff(y(x),x$2)+4*x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 30

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

\[ y(x)\to c_2 \left (x^5-x^3+x\right )+c_1 \left (x^4-x^2+1\right ) \]