87.26.22 problem 31

Internal problem ID [23856]
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 : 31
Date solved : Thursday, October 02, 2025 at 09:45:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 66

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

\[ y = \frac {c_1 \left (x -1\right )^{3} \left (1+\frac {1}{4} \left (x -1\right )+\frac {1}{40} \left (x -1\right )^{2}+\frac {1}{720} \left (x -1\right )^{3}+\frac {1}{20160} \left (x -1\right )^{4}+\frac {1}{806400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_2 \left (\ln \left (x -1\right ) \left (\left (x -1\right )^{3}+\frac {1}{4} \left (x -1\right )^{4}+\frac {1}{40} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+\left (12-6 \left (x -1\right )+3 \left (x -1\right )^{2}-\frac {5}{16} \left (x -1\right )^{4}-\frac {39}{800} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )\right )}{x -1} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 106

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

\[ y(x)\to c_2 \left (\frac {(x-1)^6}{20160}+\frac {1}{720} (x-1)^5+\frac {1}{40} (x-1)^4+\frac {1}{4} (x-1)^3+(x-1)^2\right )+c_1 \left (\frac {1}{48} (x-1)^2 (x+3) \log (x-1)-\frac {19 (x-1)^4+16 (x-1)^3-144 (x-1)^2+288 (x-1)-576}{576 (x-1)}\right ) \]

✓ Sympy. Time used: 0.351 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{1} \left (x - 1\right )^{2} \left (180 x + \left (x - 1\right )^{3} + 18 \left (x - 1\right )^{2} + 540\right )}{720} + O\left (x^{6}\right ) \]