7.15.10 problem 10

Internal problem ID [466]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 10
Date solved : Saturday, March 29, 2025 at 04:54:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 57

Order:=6; 
ode:=(1-x)^2*diff(diff(y(x),x),x)+(-2+2*x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {2}{3} x^{3}-\frac {17}{24} x^{4}-\frac {7}{10} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {5}{6} x^{3}+\frac {2}{3} x^{4}+\frac {21}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (-\frac {7 x^5}{10}-\frac {17 x^4}{24}-\frac {2 x^3}{3}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {21 x^5}{40}+\frac {2 x^4}{3}+\frac {5 x^3}{6}+x^2+x\right ) \]

✓ Sympy. Time used: 0.861 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {17 x^{4}}{24} - \frac {2 x^{3}}{3} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {2 x^{3}}{3} + \frac {5 x^{2}}{6} + x + 1\right ) + O\left (x^{6}\right ) \]