68.1.10 problem Problem 1.8(a)

Internal problem ID [14060]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.8(a)
Date solved : Wednesday, March 05, 2025 at 10:27:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{6} x^{4}-\frac {7}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}+\frac {7}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

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

✓ Sympy. Time used: 1.061 (sec). Leaf size: 61

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

\[ y{\left (x \right )} = - \frac {x^{4} r{\left (3 \right )}}{4} + \frac {11 x^{5} r{\left (3 \right )}}{20} + C_{2} \left (- \frac {x^{5}}{40} + \frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{40} - \frac {x^{3}}{8} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]