85.81.1 problem 1

Internal problem ID [22996]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. B Exercises at page 347
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:17:23 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 18

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

\[ y(x)\to -x^4-3 x^2+4 x+3 \]

✓ Sympy. Time used: 0.337 (sec). Leaf size: 41

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

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