12.9.18 problem 18

Internal problem ID [1774]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 18
Date solved : Saturday, March 29, 2025 at 11:39:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 19

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

\[ y(x)\to \frac {e^x (c_2 x+c_1)}{x} \]

✗ Sympy

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

False