67.25.12 problem 35.3 (f)

Internal problem ID [17007]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.3 (f)
Date solved : Thursday, October 02, 2025 at 01:41:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (\frac {1}{x}-\frac {1}{3}\right ) y^{\prime }+\left (\frac {1}{x}-\frac {1}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 44

Order:=6; 
ode:=diff(diff(y(x),x),x)+(1/x-1/3)*diff(y(x),x)+(1/x-1/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {11}{48} x^{2}-\frac {47}{1296} x^{3}+\frac {11}{3072} x^{4}-\frac {653}{2073600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {7}{3} x -\frac {101}{144} x^{2}+\frac {10}{81} x^{3}-\frac {6721}{497664} x^{4}+\frac {229213}{186624000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 113

ode=D[y[x],{x,2}]+(1/x-1/3)*D[y[x],x]+(1/x-1/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {653 x^5}{2073600}+\frac {11 x^4}{3072}-\frac {47 x^3}{1296}+\frac {11 x^2}{48}-x+1\right )+c_2 \left (\frac {229213 x^5}{186624000}-\frac {6721 x^4}{497664}+\frac {10 x^3}{81}-\frac {101 x^2}{144}+\left (-\frac {653 x^5}{2073600}+\frac {11 x^4}{3072}-\frac {47 x^3}{1296}+\frac {11 x^2}{48}-x+1\right ) \log (x)+\frac {7 x}{3}\right ) \]

✓ Sympy. Time used: 0.480 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-1/3 + 1/x)*Derivative(y(x), x) + (-1/4 + 1/x)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {653 x^{5}}{2073600} + \frac {11 x^{4}}{3072} - \frac {47 x^{3}}{1296} + \frac {11 x^{2}}{48} - x + 1\right ) + O\left (x^{6}\right ) \]