67.26.11 problem 36.2 (k)

Internal problem ID [17038]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (k)
Date solved : Thursday, October 02, 2025 at 01:42:12 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 62

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

\[ y = c_1 \left (x -3\right ) \left (1-\left (x -3\right )+\frac {1}{2} \left (x -3\right )^{2}-\frac {1}{6} \left (x -3\right )^{3}+\frac {1}{24} \left (x -3\right )^{4}-\frac {1}{120} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+c_2 \left (\ln \left (x -3\right ) \left (-\left (x -3\right )+\left (x -3\right )^{2}-\frac {1}{2} \left (x -3\right )^{3}+\frac {1}{6} \left (x -3\right )^{4}-\frac {1}{24} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+\left (1-\left (x -3\right )+\frac {1}{4} \left (x -3\right )^{3}-\frac {5}{36} \left (x -3\right )^{4}+\frac {13}{288} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 105

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

\[ y(x)\to c_2 \left (\frac {1}{24} (x-3)^5-\frac {1}{6} (x-3)^4+\frac {1}{2} (x-3)^3-(x-3)^2+x-3\right )+c_1 \left (\frac {1}{36} \left (-11 (x-3)^4+27 (x-3)^3-36 (x-3)^2+36\right )+\frac {1}{6} \left ((x-3)^3-3 (x-3)^2+6 (x-3)-6\right ) (x-3) \log (x-3)\right ) \]

✓ Sympy. Time used: 0.266 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{1} \left (x - 3\right ) \left (- x + \frac {\left (x - 3\right )^{4}}{24} - \frac {\left (x - 3\right )^{3}}{6} + \frac {\left (x - 3\right )^{2}}{2} + 4\right ) + O\left (x^{6}\right ) \]