46.2.11 problem 11

Internal problem ID [9548]
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 : 11
Date solved : Tuesday, September 30, 2025 at 06:20:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+5 \left (x +1\right ) y^{\prime }+\left (x^{2}-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 74

Order:=8; 
ode:=(x^2-1)*diff(diff(y(x),x),x)+5*(1+x)*diff(y(x),x)+(x^2-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{8} x^{4}-\frac {3}{10} x^{5}-\frac {17}{45} x^{6}-\frac {199}{336} x^{7}\right ) y \left (0\right )+\left (x +\frac {5}{2} x^{2}+5 x^{3}+\frac {26}{3} x^{4}+\frac {1661}{120} x^{5}+\frac {4967}{240} x^{6}+\frac {14881}{504} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 89

ode=(x^2-1)*D[y[x],{x,2}]+5*(x+1)*D[y[x],x]+(x^2-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {199 x^7}{336}-\frac {17 x^6}{45}-\frac {3 x^5}{10}-\frac {x^4}{8}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {14881 x^7}{504}+\frac {4967 x^6}{240}+\frac {1661 x^5}{120}+\frac {26 x^4}{3}+5 x^3+\frac {5 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.470 (sec). Leaf size: 117

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5*x + 5)*Derivative(y(x), x) + (x**2 - 1)*Derivative(y(x), (x, 2)) + (x**2 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {5 x^{4} r{\left (3 \right )}}{4} + \frac {23 x^{5} r{\left (3 \right )}}{10} + \frac {193 x^{6} r{\left (3 \right )}}{60} + \frac {799 x^{7} r{\left (3 \right )}}{168} + C_{2} \left (- \frac {389 x^{7}}{1008} - \frac {151 x^{6}}{480} - \frac {37 x^{5}}{240} - \frac {x^{4}}{6} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {362 x^{6}}{63} + \frac {369 x^{5}}{80} + \frac {281 x^{4}}{120} + \frac {29 x^{3}}{12} + \frac {5 x}{2} + 1\right ) + O\left (x^{8}\right ) \]