4.14.12 problem 8

Internal problem ID [1394]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 04:33:32 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{6}-\frac {\left (-1+x \right )^{4}}{24}+\frac {\left (-1+x \right )^{5}}{60}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{12}-\frac {\left (-1+x \right )^{5}}{24}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (\frac {1}{60} (x-1)^5-\frac {1}{24} (x-1)^4+\frac {1}{6} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {1}{24} (x-1)^5+\frac {1}{12} (x-1)^4-\frac {1}{6} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.258 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {\left (x - 1\right )^{4}}{24} + \frac {\left (x - 1\right )^{3}}{6} - \frac {\left (x - 1\right )^{2}}{2} + 1\right ) + C_{1} \left (x + \frac {\left (x - 1\right )^{4}}{12} - \frac {\left (x - 1\right )^{3}}{6} - 1\right ) + O\left (x^{6}\right ) \]