85.72.14 problem 2 (d)

Internal problem ID [22950]
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 316
Problem number : 2 (d)
Date solved : Thursday, October 02, 2025 at 09:16:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} {\frac {1}{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 0.263 (sec). Leaf size: 44

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

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