73.25.32 problem 35.5 (d)

Internal problem ID [15669]
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 : 35.5 (d)
Date solved : Monday, March 31, 2025 at 01:44:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -3\right )^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 52

Order:=6; 
ode:=(x-3)^2*diff(diff(y(x),x),x)+(x^2-3*x)*diff(y(x),x)-3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=3);
 

\[ y = \frac {c_1 \left (x -3\right )^{4} \left (1-\frac {1}{5} \left (x -3\right )+\frac {1}{30} \left (x -3\right )^{2}-\frac {1}{210} \left (x -3\right )^{3}+\frac {1}{1680} \left (x -3\right )^{4}-\frac {1}{15120} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+c_2 \left (-144+144 \left (x -3\right )-72 \left (x -3\right )^{2}+24 \left (x -3\right )^{3}-6 \left (x -3\right )^{4}+\frac {6}{5} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )}{\left (x -3\right )^{3}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 81

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

\[ y(x)\to c_1 \left (\frac {x-3}{24}+\frac {1}{2 (x-3)}-\frac {1}{(x-3)^2}+\frac {1}{(x-3)^3}-\frac {1}{6}\right )+c_2 \left (\frac {(x-3)^5}{1680}-\frac {1}{210} (x-3)^4+\frac {1}{30} (x-3)^3-\frac {1}{5} (x-3)^2+x-3\right ) \]

✓ Sympy. Time used: 1.126 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = \frac {C_{2} \left (x - 3\right ) \left (- 336 x + \left (x - 3\right )^{4} - 8 \left (x - 3\right )^{3} + 56 \left (x - 3\right )^{2} + 2688\right )}{1680} + \frac {C_{1} \left (- x - \frac {\left (x - 3\right )^{3}}{6} + \frac {\left (x - 3\right )^{2}}{2} + 4\right )}{\left (x - 3\right )^{3}} + O\left (x^{6}\right ) \]