43.1.3 problem 7.2.3

Internal problem ID [6848]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.2.1 Exercises. page 290
Problem number : 7.2.3
Date solved : Sunday, March 30, 2025 at 11:24:24 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 69

Order:=6; 
ode:=diff(diff(y(x),x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{24}+\frac {\left (x -1\right )^{5}}{30}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{6}+\frac {\left (x -1\right )^{4}}{12}+\frac {\left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

ode=D[y[x],{x,2}]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{30} (x-1)^5+\frac {1}{24} (x-1)^4+\frac {1}{6} (x-1)^3+\frac {1}{2} (x-1)^2+1\right )+c_2 \left (\frac {1}{120} (x-1)^5+\frac {1}{12} (x-1)^4+\frac {1}{6} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.868 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 1\right )^{4}}{24} + \frac {\left (x - 1\right )^{3}}{6} + \frac {\left (x - 1\right )^{2}}{2} + 1\right ) + C_{1} \left (x + \frac {\left (x - 1\right )^{4}}{12} + \frac {\left (x - 1\right )^{3}}{6} - 1\right ) + O\left (x^{6}\right ) \]