82.3.7 problem 25-7

Internal problem ID [21790]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 25. Power series about a singular point. Page 762
Problem number : 25-7
Date solved : Thursday, October 02, 2025 at 08:02:14 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 4 x y^{\prime \prime }+2 y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

Order:=6; 
ode:=4*x*diff(diff(y(x),x),x)+2*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{6} x +\frac {1}{120} x^{2}+\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}+\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{2} x +\frac {1}{24} x^{2}+\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}+\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 85

ode=4*x*D[y[x],{x,2}]+2*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{39916800}+\frac {x^4}{362880}+\frac {x^3}{5040}+\frac {x^2}{120}+\frac {x}{6}+1\right )+c_2 \left (\frac {x^5}{3628800}+\frac {x^4}{40320}+\frac {x^3}{720}+\frac {x^2}{24}+\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.281 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) - y(x) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{3628800} + \frac {x^{4}}{40320} + \frac {x^{3}}{720} + \frac {x^{2}}{24} + \frac {x}{2} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{362880} + \frac {x^{3}}{5040} + \frac {x^{2}}{120} + \frac {x}{6} + 1\right ) + O\left (x^{6}\right ) \]