67.25.10 problem 35.3 (d)

Internal problem ID [17005]
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 (d)
Date solved : Thursday, October 02, 2025 at 01:41:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x -3}+\frac {y}{x -4}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 42

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

\[ y = \left (c_2 \ln \left (x -3\right )+c_1 \right ) \left (1+\frac {1}{4} \left (x -3\right )^{2}+\frac {1}{9} \left (x -3\right )^{3}+\frac {5}{64} \left (x -3\right )^{4}+\frac {49}{900} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+\left (-\frac {1}{4} \left (x -3\right )^{2}-\frac {2}{27} \left (x -3\right )^{3}-\frac {7}{128} \left (x -3\right )^{4}-\frac {469}{13500} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 128

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

\[ y(x)\to c_1 \left (\frac {49}{900} (x-3)^5+\frac {5}{64} (x-3)^4+\frac {1}{9} (x-3)^3+\frac {1}{4} (x-3)^2+1\right )+c_2 \left (-\frac {469 (x-3)^5}{13500}-\frac {7}{128} (x-3)^4-\frac {2}{27} (x-3)^3-\frac {1}{4} (x-3)^2+\left (\frac {49}{900} (x-3)^5+\frac {5}{64} (x-3)^4+\frac {1}{9} (x-3)^3+\frac {1}{4} (x-3)^2+1\right ) \log (x-3)\right ) \]

✓ Sympy. Time used: 0.356 (sec). Leaf size: 7

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

\[ y{\left (x \right )} = C_{1} + O\left (x^{6}\right ) \]