50.22.13 problem 2(e)

Internal problem ID [8156]
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 : 2(e)
Date solved : Sunday, March 30, 2025 at 12:46:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 40

Order:=8; 
ode:=2*x*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{6} x -\frac {1}{120} x^{2}-\frac {1}{1680} x^{3}-\frac {1}{24192} x^{4}-\frac {1}{380160} x^{5}-\frac {1}{6589440} x^{6}-\frac {1}{125798400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {17333 x^7}{48432384000}-\frac {34817 x^6}{691891200}-\frac {1171 x^5}{4435200}+\frac {121 x^4}{40320}+\frac {37 x^3}{1680}-\frac {3 x^2}{40}-\frac {x}{2}+1\right )+c_2 \left (\frac {4 x^7}{143325}-\frac {x^6}{8400}-\frac {19 x^5}{6300}-\frac {x^4}{840}+\frac {2 x^3}{15}+\frac {x^2}{6}-2 x+1\right ) \]

✓ Sympy. Time used: 0.819 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (1 - x\right ) + C_{1} \sqrt {x} \left (- \frac {x^{6}}{6589440} - \frac {x^{5}}{380160} - \frac {x^{4}}{24192} - \frac {x^{3}}{1680} - \frac {x^{2}}{120} - \frac {x}{6} + 1\right ) + O\left (x^{8}\right ) \]