6.12.17 problem 19

Internal problem ID [1871]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:20:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 41

Order:=6; 
ode:=(2*x^2-8*x+11)*diff(diff(y(x),x),x)-16*(x-2)*diff(y(x),x)+36*y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = \left (1-6 \left (x -2\right )^{2}+\frac {4 \left (x -2\right )^{4}}{3}\right ) y \left (2\right )+\left (x -2-\frac {10 \left (x -2\right )^{3}}{9}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=(11-8*x+2*x^2)*D[y[x],{x,2}]-16*(x-2)*D[y[x],x]+36*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (\frac {4}{3} (x-2)^4-6 (x-2)^2+1\right )+c_2 \left (-\frac {10}{9} (x-2)^3+x-2\right ) \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((32 - 16*x)*Derivative(y(x), x) + (2*x**2 - 8*x + 11)*Derivative(y(x), (x, 2)) + 36*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {4 \left (x - 2\right )^{4}}{3} - 6 \left (x - 2\right )^{2} + 1\right ) + C_{1} \left (x - \frac {10 \left (x - 2\right )^{3}}{9} - 2\right ) + O\left (x^{6}\right ) \]