64.14.13 problem 13

Internal problem ID [13572]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 05:53:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+2 y x&=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 )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

Order:=6; 
dsolve([(x^2+1)*diff(y(x),x$2)+x*diff(y(x),x)+2*x*y(x)=0,y(0) = 2, D(y)(0) = 3],y(x),type='series',x=0);
 

\[ y = 2+3 x -\frac {7}{6} x^{3}-\frac {1}{2} x^{4}+\frac {21}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 29

AsymptoticDSolveValue[{(x^2+1)*D[y[x],{x,2}]+x*D[y[x],x]+2*x*y[x]==0,{y[0]==2,Derivative[1][y][0] ==3}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {21 x^5}{40}-\frac {x^4}{2}-\frac {7 x^3}{6}+3 x+2 \]