85.72.12 problem 2 (b)

Internal problem ID [22948]
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 (b)
Date solved : Thursday, October 02, 2025 at 09:16:52 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 69

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

\[ y = \left (1+\frac {\left (x -2\right )^{2}}{8}-\frac {\left (x -2\right )^{3}}{24}+\frac {7 \left (x -2\right )^{4}}{384}-\frac {\left (x -2\right )^{5}}{120}\right ) y \left (2\right )+\left (x -2+\frac {\left (x -2\right )^{3}}{24}-\frac {\left (x -2\right )^{4}}{48}+\frac {19 \left (x -2\right )^{5}}{1920}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_1 \left (-\frac {1}{120} (x-2)^5+\frac {7}{384} (x-2)^4-\frac {1}{24} (x-2)^3+\frac {1}{8} (x-2)^2+1\right )+c_2 \left (\frac {19 (x-2)^5}{1920}-\frac {1}{48} (x-2)^4+\frac {1}{24} (x-2)^3+x-2\right ) \]

✓ Sympy. Time used: 0.259 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{2} \left (\frac {7 \left (x - 2\right )^{4}}{384} - \frac {\left (x - 2\right )^{3}}{24} + \frac {\left (x - 2\right )^{2}}{8} + 1\right ) + C_{1} \left (x - \frac {\left (x - 2\right )^{4}}{48} + \frac {\left (x - 2\right )^{3}}{24} - 2\right ) + O\left (x^{6}\right ) \]