82.3.10 problem 25-10

Internal problem ID [21793]
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 : 25-10
Date solved : Thursday, October 02, 2025 at 08:02:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 18 x^{2} y^{\prime \prime }+3 x \left (5+x \right ) y^{\prime }-\left (10 x +1\right ) 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:=18*x^2*diff(diff(y(x),x),x)+3*x*(x+5)*diff(y(x),x)-(10*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{45} x^{2}+\frac {1}{2835} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {7}{6} x +\frac {35}{216} x^{2}+\frac {7}{1296} x^{3}+\frac {1}{31104} x^{4}-\frac {1}{8398080} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{6}}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 76

ode=18*x^2*D[y[x],{x,2}]+3*x*(x+5)*D[y[x],x]-(10*x+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^3}{2835}+\frac {x^2}{45}+\frac {x}{3}+1\right )+\frac {c_2 \left (-\frac {x^5}{8398080}+\frac {x^4}{31104}+\frac {7 x^3}{1296}+\frac {35 x^2}{216}+\frac {7 x}{6}+1\right )}{\sqrt [6]{x}} \]

✓ Sympy. Time used: 0.406 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(18*x**2*Derivative(y(x), (x, 2)) + 3*x*(x + 5)*Derivative(y(x), x) - (10*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {x^{3}}{2835} + \frac {x^{2}}{45} + \frac {x}{3} + 1\right ) + \frac {C_{1} \left (- \frac {x^{5}}{8398080} + \frac {x^{4}}{31104} + \frac {7 x^{3}}{1296} + \frac {35 x^{2}}{216} + \frac {7 x}{6} + 1\right )}{\sqrt [6]{x}} + O\left (x^{6}\right ) \]