82.2.14 problem 24-13

Internal problem ID [21775]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-13
Date solved : Thursday, October 02, 2025 at 08:02:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+3 x y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

Order:=6; 
ode:=(x^2-1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+x*y(x) = 0; 
ic:=[y(0) = 4, D(y)(0) = 6]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 4+6 x +\frac {11}{3} x^{3}+\frac {1}{2} x^{4}+\frac {11}{4} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

ode=(x^2-1)*D[y[x],{x,2}]+3*x*D[y[x],x]+x*y[x]==0; 
ic={y[0]==4,Derivative[1][y][0] ==6}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {11 x^5}{4}+\frac {x^4}{2}+\frac {11 x^3}{3}+6 x+4 \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + 3*x*Derivative(y(x), x) + (x**2 - 1)*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 4, Subs(Derivative(y(x), x), x, 0): 6} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{8} + \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {3 x^{4}}{8} + \frac {x^{3}}{12} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]