87.26.19 problem 28

Internal problem ID [23853]
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 253
Problem number : 28
Date solved : Thursday, October 02, 2025 at 09:45:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 38

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

\[ y = \left (3 \left (x -1\right )-\frac {1}{4} \left (x -1\right )^{2}-\frac {1}{36} \left (x -1\right )^{3}-\frac {1}{288} \left (x -1\right )^{4}-\frac {1}{2400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) c_2 +\left (1-\left (x -1\right )+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) \left (c_2 \ln \left (x -1\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 69

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

\[ y(x)\to c_1 (2-x)+c_2 \left (-\frac {(x-1)^5}{2400}-\frac {1}{288} (x-1)^4-\frac {1}{36} (x-1)^3-\frac {1}{4} (x-1)^2+2 (x-1)+x+(2-x) \log (x-1)-1\right ) \]

✓ Sympy. Time used: 0.293 (sec). Leaf size: 10

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

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