46.2.4 problem 4

Internal problem ID [9541]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 06:20:24 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: 0.021 (sec). Leaf size: 64

Order:=8; 
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);
 

\[ y = c_1 \,x^{2} \left (1+\frac {1}{8} x^{2}+\frac {1}{5} x^{3}+\frac {49}{192} x^{4}+\frac {423}{1400} x^{5}+\frac {15941}{46080} x^{6}+\frac {30511}{78400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{2}-\frac {1}{8} x^{4}-\frac {1}{5} x^{5}-\frac {49}{192} x^{6}-\frac {423}{1400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-2 x^{3}-\frac {45}{32} x^{4}-\frac {34}{25} x^{5}-\frac {1673}{1152} x^{6}-\frac {313337}{196000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 107

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,0,7}]
 

\[ y(x)\to c_1 \left (\frac {\left (245 x^4+192 x^3+120 x^2+960\right ) x^2 \log (x)}{1920}+\frac {-25025 x^6-16416 x^5-2250 x^4+28800 x^3-180000 x^2+28800}{28800}\right )+c_2 \left (\frac {15941 x^8}{46080}+\frac {423 x^7}{1400}+\frac {49 x^6}{192}+\frac {x^5}{5}+\frac {x^4}{8}+x^2\right ) \]

✓ Sympy. Time used: 0.422 (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=8)
 

\[ y{\left (x \right )} = C_{2} x^{2} + C_{1} + O\left (x^{8}\right ) \]