87.24.6 problem 6

Internal problem ID [23788]
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 218
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:45:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \,x^{3} \left (1+\frac {1}{4} x +\frac {1}{40} x^{2}+\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}+\frac {1}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x^{3}+\frac {1}{4} x^{4}+\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-6 x +3 x^{2}-\frac {5}{16} x^{4}-\frac {39}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 82

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

\[ y(x)\to c_1 \left (\frac {1}{48} x^2 (x+4) \log (x)-\frac {19 x^4+16 x^3-144 x^2+288 x-576}{576 x}\right )+c_2 \left (\frac {x^6}{20160}+\frac {x^5}{720}+\frac {x^4}{40}+\frac {x^3}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.257 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {x^{3}}{720} + \frac {x^{2}}{40} + \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]