85.73.9 problem 9

Internal problem ID [22962]
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 316
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:16:59 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 40

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

\[ y = \left (-\frac {13}{2} \left (x -1\right )-\frac {47}{8} \left (x -1\right )^{2}-\frac {5}{12} \left (x -1\right )^{3}+\frac {5}{64} \left (x -1\right )^{4}-\frac {7}{320} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) c_2 +\left (1+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) \left (c_2 \ln \left (x -1\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 98

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

\[ y(x)\to c_1 \left (\frac {3}{2} (x-1)^2+3 (x-1)+1\right )+c_2 \left (-\frac {7}{320} (x-1)^5+\frac {5}{64} (x-1)^4-\frac {5}{12} (x-1)^3-\frac {47}{8} (x-1)^2-6 (x-1)+\frac {1-x}{2}+\left (\frac {3}{2} (x-1)^2+3 (x-1)+1\right ) \log (x-1)\right ) \]

✓ Sympy. Time used: 2.543 (sec). Leaf size: 534

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

\[ \text {Solution too large to show} \]