87.14.12 problem 12

Internal problem ID [23531]
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 2. Linear differential equations. Exercise at page 109
Problem number : 12
Date solved : Thursday, October 02, 2025 at 09:42:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \left (\frac {1}{x}-\frac {1}{x^{2}}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

ode:=(x-2)*x*diff(diff(y(x),x),x)-2*(x^2-3*x+3)*diff(y(x),x)+(x^2-4*x+6)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (c_1 +\frac {c_2 \left (x -1\right )}{x^{2}}\right ) \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 27

ode=(x-2)*x*D[y[x],{x,2}]-2*(x^2-3*x+3)*D[y[x],x]+(x^2-4*x+6)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^x \left (c_1 (x-2)^2+c_2 (x-1)\right )}{x^2} \end{align*}

✗ Sympy

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

False