84.33.1 problem 20.1

Internal problem ID [22318]
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. Solved problems. Page 109
Problem number : 20.1
Date solved : Thursday, October 02, 2025 at 08:37:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} y^{\prime \prime }+10 x y^{\prime }+\left (x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 48

Order:=6; 
ode:=8*x^2*diff(diff(y(x),x),x)+10*x*diff(y(x),x)+(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{3}/{4}} \left (1-\frac {1}{14} x +\frac {1}{616} x^{2}-\frac {1}{55440} x^{3}+\frac {1}{8426880} x^{4}-\frac {1}{1938182400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {1}{2} x +\frac {1}{40} x^{2}-\frac {1}{2160} x^{3}+\frac {1}{224640} x^{4}-\frac {1}{38188800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 90

ode=8*x^2*D[y[x],{x,2}]+10*x*D[y[x],x]+(x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {x^5}{1938182400}+\frac {x^4}{8426880}-\frac {x^3}{55440}+\frac {x^2}{616}-\frac {x}{14}+1\right )+\frac {c_2 \left (-\frac {x^5}{38188800}+\frac {x^4}{224640}-\frac {x^3}{2160}+\frac {x^2}{40}-\frac {x}{2}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.339 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*Derivative(y(x), (x, 2)) + 10*x*Derivative(y(x), x) + (x - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [4]{x} \left (\frac {x^{4}}{8426880} - \frac {x^{3}}{55440} + \frac {x^{2}}{616} - \frac {x}{14} + 1\right ) + \frac {C_{1} \left (- \frac {x^{5}}{38188800} + \frac {x^{4}}{224640} - \frac {x^{3}}{2160} + \frac {x^{2}}{40} - \frac {x}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]