39.2.4 problem 4

Internal problem ID [8513]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:38:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}+\frac {y}{\left (x -1\right )^{3}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 52

Order:=6; 
ode:=diff(diff(y(x),x),x)-1/x*diff(y(x),x)+1/(x-1)^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1+\frac {1}{8} x^{2}+\frac {1}{5} x^{3}+\frac {49}{192} x^{4}+\frac {423}{1400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{2}-\frac {1}{8} x^{4}-\frac {1}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x^{3}-\frac {45}{32} x^{4}-\frac {34}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 71

ode=D[y[x],{x,2}]-1/x*D[y[x],x]+1/(x-1)^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{16} \left (x^2+8\right ) x^2 \log (x)+\frac {1}{64} \left (-5 x^4+64 x^3-400 x^2+64\right )\right )+c_2 \left (\frac {49 x^6}{192}+\frac {x^5}{5}+\frac {x^4}{8}+x^2\right ) \]

✓ Sympy. Time used: 0.424 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + y(x)/(x - 1)**3 - Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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