87.25.4 problem 4

Internal problem ID [23807]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 232
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:45:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ y^{\prime }\left (2\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 13

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

\[ y = -3 \left (x -2\right )^{2}+1 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 12

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

\[ y(x)\to 1-3 (x-2)^2 \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 29

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

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