6.13.10 problem 10

Internal problem ID [1901]
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 : 10
Date solved : Tuesday, September 30, 2025 at 05:21:04 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

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

\[ y = 2-\left (x -1\right )-\frac {1}{2} \left (x -1\right )^{2}+\frac {5}{3} \left (x -1\right )^{3}-\frac {19}{12} \left (x -1\right )^{4}+\frac {7}{30} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 44

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

\[ y(x)\to \frac {7}{30} (x-1)^5-\frac {19}{12} (x-1)^4+\frac {5}{3} (x-1)^3-\frac {1}{2} (x-1)^2-x+3 \]

✓ Sympy. Time used: 0.321 (sec). Leaf size: 56

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

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