87.24.5 problem 5

Internal problem ID [23787]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 218
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:45:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {2}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (-\frac {2 x^5}{15}-\frac {x^4}{4}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {x^5}{30}-\frac {x^4}{8}-\frac {x^3}{3}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.338 (sec). Leaf size: 39

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

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