87.26.17 problem 26

Internal problem ID [23851]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 26
Date solved : Thursday, October 02, 2025 at 09:45:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 76

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-1/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {4 \left (x -1\right )^{2}}{9}+\frac {\left (x -1\right )^{3}}{9}-\frac {10 \left (x -1\right )^{4}}{243}+\frac {229 \left (x -1\right )^{5}}{4860}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{2}+\frac {5 \left (x -1\right )^{3}}{27}-\frac {7 \left (x -1\right )^{4}}{36}+\frac {439 \left (x -1\right )^{5}}{2430}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {229 (x-1)^5}{4860}-\frac {10}{243} (x-1)^4+\frac {1}{9} (x-1)^3-\frac {4}{9} (x-1)^2+1\right )+c_2 \left (\frac {439 (x-1)^5}{2430}-\frac {7}{36} (x-1)^4+\frac {5}{27} (x-1)^3-\frac {1}{2} (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 1/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = - \frac {5 \left (x - 1\right )^{4} r{\left (3 \right )}}{4} + \frac {113 \left (x - 1\right )^{5} r{\left (3 \right )}}{90} + C_{2} \left (x - \frac {7 \left (x - 1\right )^{5}}{135} + \frac {\left (x - 1\right )^{4}}{27} - \frac {\left (x - 1\right )^{2}}{2} - 1\right ) + C_{1} \left (- \frac {398 \left (x - 1\right )^{5}}{1215} + \frac {293 \left (x - 1\right )^{4}}{972} - \frac {17 \left (x - 1\right )^{2}}{18} + 1\right ) + O\left (x^{6}\right ) \]