85.80.1 problem 2

Internal problem ID [22995]
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. A Exercises at page 346
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:17:22 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*} {\frac {1}{2}} \end{align*}

With initial conditions

\begin{align*} y \left (\frac {1}{2}\right )&=10 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 41

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+6*y(x) = 0; 
ic:=[y(1/2) = 10]; 
dsolve([ode,op(ic)],y(x),type='series',x=1/2);
 

\[ y = 10+y^{\prime }\left (\frac {1}{2}\right ) \left (x -\frac {1}{2}\right )+\left (\frac {2 y^{\prime }\left (\frac {1}{2}\right )}{3}-40\right ) \left (x -\frac {1}{2}\right )^{2}+\left (-\frac {8 y^{\prime }\left (\frac {1}{2}\right )}{27}-\frac {320}{9}\right ) \left (x -\frac {1}{2}\right )^{3}+\left (-\frac {8 y^{\prime }\left (\frac {1}{2}\right )}{27}-\frac {320}{9}\right ) \left (x -\frac {1}{2}\right )^{4}+\left (-\frac {176 y^{\prime }\left (\frac {1}{2}\right )}{405}-\frac {1408}{27}\right ) \left (x -\frac {1}{2}\right )^{5}+\operatorname {O}\left (\left (x -\frac {1}{2}\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 124

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+6*y[x]==0; 
ic={y[1/2]==10}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1/2,5}]
 

\[ y(x)\to -\frac {11264 (80+c_1) \left (x-\frac {1}{2}\right )^5}{1215 (12+\log (3))}-\frac {512 (80+c_1) \left (x-\frac {1}{2}\right )^4}{81 (12+\log (3))}-\frac {512 (80+c_1) \left (x-\frac {1}{2}\right )^3}{81 (12+\log (3))}+\frac {4 \left (x-\frac {1}{2}\right )^2 \left (-680-135 \log (4)-270 \log \left (\frac {3}{2}\right )+32 c_1\right )}{9 (12+\log (3))}+\frac {4 \left (x-\frac {1}{2}\right ) (200-5 \log (387420489)+16 c_1)}{36+\log (27)}+10 \]

✓ Sympy. Time used: 0.375 (sec). Leaf size: 75

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 = {y(1/2): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1/2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {8 \left (x - \frac {1}{2}\right )^{4}}{27} - \frac {8 \left (x - \frac {1}{2}\right )^{3}}{27} + \frac {2 \left (x - \frac {1}{2}\right )^{2}}{3} - \frac {1}{2}\right ) + C_{1} \left (- \frac {32 \left (x - \frac {1}{2}\right )^{4}}{9} - \frac {32 \left (x - \frac {1}{2}\right )^{3}}{9} - 4 \left (x - \frac {1}{2}\right )^{2} + 1\right ) + O\left (x^{6}\right ) \]