44.22.8 problem 1(h)

Internal problem ID [9437]
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(h)
Date solved : Tuesday, September 30, 2025 at 06:18:47 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 79

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

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {3}{8} x^{4}+\frac {11}{30} x^{5}+\frac {53}{144} x^{6}+\frac {103}{280} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{8} x^{4}+\frac {19}{30} x^{5}+\frac {91}{144} x^{6}+\frac {177}{280} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 98

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

\[ y(x)\to c_1 \left (\frac {103 x^7}{280}+\frac {53 x^6}{144}+\frac {11 x^5}{30}+\frac {3 x^4}{8}+\frac {x^3}{3}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {177 x^7}{280}+\frac {91 x^6}{144}+\frac {19 x^5}{30}+\frac {5 x^4}{8}+\frac {2 x^3}{3}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.344 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {53 x^{6}}{144} + \frac {11 x^{5}}{30} + \frac {3 x^{4}}{8} + \frac {x^{3}}{3} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {91 x^{5}}{144} + \frac {19 x^{4}}{30} + \frac {5 x^{3}}{8} + \frac {2 x^{2}}{3} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]