64.14.12 problem 12

Internal problem ID [13571]
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 : 12
Date solved : Tuesday, January 28, 2025 at 05:53:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

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

\[ y = x +\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 19

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

\[ y(x)\to -\frac {x^5}{120}+\frac {x^3}{6}+x \]