43.14.2 problem 2

Internal problem ID [8971]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 124
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:00:45 PM
CAS classification : [[_Emden, _Fowler]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 12

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

\[ y = x \left (\ln \left (x \right ) c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 15

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

\begin{align*} y(x)&\to x (c_2 \log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = x \left (C_{1} + C_{2} \log {\left (x \right )}\right ) \]