6.12.21 problem 23

Internal problem ID [1875]
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 : 23
Date solved : Tuesday, September 30, 2025 at 05:20:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}-6 x +5\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+12 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: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=(3*x^2-6*x+5)*diff(diff(y(x),x),x)+(x-1)*diff(y(x),x)+12*y(x) = 0; 
ic:=[y(1) = -1, D(y)(1) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = -1+\left (x -1\right )+3 \left (x -1\right )^{2}-\frac {13}{12} \left (x -1\right )^{3}-\frac {5}{2} \left (x -1\right )^{4}+\frac {143}{160} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

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

ode=(5-6*x+3*x^2)*D[y[x],{x,2}]+(x-1)*D[y[x],x]+12*y[x]==0; 
ic={y[1]==-1,Derivative[1][y][1]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to \frac {143}{160} (x-1)^5-\frac {5}{2} (x-1)^4-\frac {13}{12} (x-1)^3+3 (x-1)^2+x-2 \]

✓ Sympy. Time used: 0.299 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*Derivative(y(x), x) + (3*x**2 - 6*x + 5)*Derivative(y(x), (x, 2)) + 12*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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {5 \left (x - 1\right )^{4}}{2} - 3 \left (x - 1\right )^{2} + 1\right ) + C_{1} \left (x - \frac {13 \left (x - 1\right )^{3}}{12} - 1\right ) + O\left (x^{6}\right ) \]