84.33.6 problem 20.6

Internal problem ID [22323]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Solved problems. Page 109
Problem number : 20.6
Date solved : Thursday, October 02, 2025 at 08:37:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 60

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

\[ y = x \left (c_1 x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}+\frac {13}{288} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{6} x^2 \left (x^3-3 x^2+6 x-6\right ) \log (x)-\frac {1}{36} x \left (11 x^4-27 x^3+36 x^2-36\right )\right )+c_2 \left (\frac {x^6}{24}-\frac {x^5}{6}+\frac {x^4}{2}-x^3+x^2\right ) \]

✓ Sympy. Time used: 0.268 (sec). Leaf size: 24

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

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