84.34.2 problem 20.15

Internal problem ID [22329]
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 : 20.15
Date solved : Thursday, October 02, 2025 at 08:37:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 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: 0.007 (sec). Leaf size: 45

Order:=6; 
ode:=2*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);
 

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1-\frac {1}{3} x +\frac {1}{15} x^{2}-\frac {1}{105} x^{3}+\frac {1}{945} x^{4}-\frac {1}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (-\frac {x^5}{10395}+\frac {x^4}{945}-\frac {x^3}{105}+\frac {x^2}{15}-\frac {x}{3}+1\right )+c_2 \sqrt {x} \left (-\frac {x^5}{3840}+\frac {x^4}{384}-\frac {x^3}{48}+\frac {x^2}{8}-\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.335 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*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)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{945} - \frac {x^{3}}{105} + \frac {x^{2}}{15} - \frac {x}{3} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{384} - \frac {x^{3}}{48} + \frac {x^{2}}{8} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]