84.34.1 problem 20.14

Internal problem ID [22328]
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 : 20.14
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 }-x y^{\prime }+\left (1-x \right ) 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*diff(y(x),x)+(1-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+x +\frac {1}{6} x^{2}+\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}+\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {1}{3} x +\frac {1}{30} x^{2}+\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}+\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 82

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

\[ y(x)\to c_1 x \left (\frac {x^5}{1247400}+\frac {x^4}{22680}+\frac {x^3}{630}+\frac {x^2}{30}+\frac {x}{3}+1\right )+c_2 \sqrt {x} \left (\frac {x^5}{113400}+\frac {x^4}{2520}+\frac {x^3}{90}+\frac {x^2}{6}+x+1\right ) \]

✓ Sympy. Time used: 0.342 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{22680} + \frac {x^{3}}{630} + \frac {x^{2}}{30} + \frac {x}{3} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{2520} + \frac {x^{3}}{90} + \frac {x^{2}}{6} + x + 1\right ) + O\left (x^{6}\right ) \]