68.17.20 problem 19 (c)

Internal problem ID [17838]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 19 (c)
Date solved : Thursday, October 02, 2025 at 02:28:45 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 81

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

\[ y = \left (1-\frac {k \left (k +1\right ) x^{2}}{2}+\frac {k \left (k^{3}+2 k^{2}-5 k -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (k^{2}+k -2\right ) x^{3}}{6}+\frac {\left (k^{4}+2 k^{3}-13 k^{2}-14 k +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 120

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

\[ y(x)\to c_2 \left (\frac {1}{120} \left (k^2+k\right )^2 x^5+\frac {7}{60} \left (-k^2-k\right ) x^5+\frac {1}{6} \left (-k^2-k\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (k^2+k\right )^2 x^4+\frac {1}{4} \left (-k^2-k\right ) x^4+\frac {1}{2} \left (-k^2-k\right ) x^2+1\right ) \]

✓ Sympy. Time used: 0.452 (sec). Leaf size: 83

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

\[ y{\left (x \right )} = C_{2} \left (\frac {k^{4} x^{4}}{24} + \frac {k^{3} x^{4}}{12} - \frac {5 k^{2} x^{4}}{24} - \frac {k^{2} x^{2}}{2} - \frac {k x^{4}}{4} - \frac {k x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {k^{2} x^{2}}{6} - \frac {k x^{2}}{6} + \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]