6.12.16 problem 18

Internal problem ID [1870]
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 : 18
Date solved : Tuesday, September 30, 2025 at 05:20:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}-4 x +1\right ) y^{\prime \prime }+10 \left (x -1\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 48

Order:=6; 
ode:=(2*x^2-4*x+1)*diff(diff(y(x),x),x)+10*(x-1)*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1+3 \left (x -1\right )^{2}+\frac {15 \left (x -1\right )^{4}}{2}\right ) y \left (1\right )+\left (x -1+\frac {8 \left (x -1\right )^{3}}{3}+\frac {32 \left (x -1\right )^{5}}{5}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

ode=(1-4*x+2*x^2)*D[y[x],{x,2}]+10*(x-1)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to c_1 \left (\frac {15}{2} (x-1)^4+3 (x-1)^2+1\right )+c_2 \left (\frac {32}{5} (x-1)^5+\frac {8}{3} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.287 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((10*x - 10)*Derivative(y(x), x) + (2*x**2 - 4*x + 1)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {15 \left (x - 1\right )^{4}}{2} + 3 \left (x - 1\right )^{2} + 1\right ) + C_{1} \left (x + \frac {8 \left (x - 1\right )^{3}}{3} - 1\right ) + O\left (x^{6}\right ) \]