87.24.8 problem 8

Internal problem ID [23790]
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 : 8
Date solved : Thursday, October 02, 2025 at 09:45:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -1\right )^{4} y^{\prime \prime }-y x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 44

Order:=6; 
ode:=(x-1)^4*diff(diff(y(x),x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{3} x^{4}+\frac {1}{2} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}+\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

ode=(x-1)^4*D[y[x],{x,2}]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{5}+\frac {x^4}{12}+x\right )+c_1 \left (\frac {x^5}{2}+\frac {x^4}{3}+\frac {x^3}{6}+1\right ) \]

✓ Sympy. Time used: 0.370 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + (x - 1)**4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = 2 x^{4} r{\left (3 \right )} + 3 x^{5} r{\left (3 \right )} + C_{2} \left (- \frac {39 x^{5}}{80} - \frac {x^{4}}{4} + \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{4}}{5} + \frac {x^{3}}{12} + 1\right ) + O\left (x^{6}\right ) \]