84.34.6 problem 20.19

Internal problem ID [22333]
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. Supplementary problems
Problem number : 20.19
Date solved : Thursday, October 02, 2025 at 08:37:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 45

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

\[ y = c_1 x \left (1+\frac {1}{3} x +\frac {1}{12} x^{2}+\frac {1}{60} x^{3}+\frac {1}{360} x^{4}+\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^5}{360}+\frac {x^4}{60}+\frac {x^3}{12}+\frac {x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.348 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{360} + \frac {x^{3}}{60} + \frac {x^{2}}{12} + \frac {x}{3} + 1\right ) + \frac {C_{1} \left (x + 1\right )}{x} + O\left (x^{6}\right ) \]