12.12.11 problem 11

Internal problem ID [1865]
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 : 11
Date solved : Monday, January 27, 2025 at 05:37:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

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

\[ y = 2-x -x^{2}+\frac {1}{3} x^{3}+\frac {5}{12} x^{4}-\frac {1}{6} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {x^5}{6}+\frac {5 x^4}{12}+\frac {x^3}{3}-x^2-x+2 \]