50.18.2 problem 1(b)

Internal problem ID [8096]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 1(b)
Date solved : Sunday, March 30, 2025 at 12:45:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 74

Order:=8; 
ode:=diff(diff(y(x),x),x)-diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{630} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}-\frac {1}{30} x^{5}-\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 91

ode=D[y[x],{x,2}]-D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^7}{630}+\frac {x^6}{240}-\frac {x^5}{120}-\frac {x^4}{24}-\frac {x^3}{6}+1\right )+c_2 \left (-\frac {x^7}{1680}-\frac {x^6}{90}-\frac {x^5}{30}-\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.856 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{240} - \frac {x^{5}}{120} - \frac {x^{4}}{24} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (- \frac {x^{5}}{90} - \frac {x^{4}}{30} - \frac {x^{3}}{24} + \frac {x^{2}}{6} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]