44.22.7 problem 1(g)

Internal problem ID [9436]
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. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 1(g)
Date solved : Tuesday, September 30, 2025 at 06:18:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 74

Order:=8; 
ode:=diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}+\frac {37}{5040} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {47}{720} x^{6}+\frac {19}{630} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

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

ode=D[y[x],{x,2}]-(x+1)*D[y[x],x]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {37 x^7}{5040}+\frac {x^6}{60}+\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}+1\right )+c_2 \left (\frac {19 x^7}{630}+\frac {47 x^6}{720}+\frac {x^5}{8}+\frac {x^4}{4}+\frac {x^3}{3}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.301 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) - (x + 1)*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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{60} + \frac {x^{5}}{30} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {47 x^{5}}{720} + \frac {x^{4}}{8} + \frac {x^{3}}{4} + \frac {x^{2}}{3} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]