6.13.8 problem 8

Internal problem ID [1899]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:21:03 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}-3 x +2\right ) y^{\prime \prime }-\left (4-6 x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ y^{\prime }\left (1\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 18

Order:=6; 
ode:=(2*x^2-3*x+2)*diff(diff(y(x),x),x)-(4-6*x)*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(1) = 1, D(y)(1) = -1]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = 1-\left (x -1\right )+\frac {4}{3} \left (x -1\right )^{3}-\frac {4}{3} \left (x -1\right )^{4}-\frac {4}{5} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

ode=(2-3*x+2*x^2)*D[y[x],{x,2}]-(4-6*x)*D[y[x],x]+2*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1]==-1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to -\frac {4}{5} (x-1)^5-\frac {4}{3} (x-1)^4+\frac {4}{3} (x-1)^3-x+2 \]

✓ Sympy. Time used: 0.322 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((6*x - 4)*Derivative(y(x), x) + (2*x**2 - 3*x + 2)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): -1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {11 \left (x - 1\right )^{4}}{6} - \frac {\left (x - 1\right )^{3}}{3} - \left (x - 1\right )^{2} - 1\right ) + C_{1} \left (\frac {\left (x - 1\right )^{4}}{2} + \left (x - 1\right )^{3} - \left (x - 1\right )^{2} + 1\right ) + O\left (x^{6}\right ) \]