87.13.15 problem 16

Internal problem ID [23498]
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 100
Problem number : 16
Date solved : Thursday, October 02, 2025 at 09:42:27 PM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 11

ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 0; 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x -x \ln \left (x \right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 12

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0; 
ic={y[1]==1,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x-x \log (x) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 8

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 = {y(1): 1, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (1 - \log {\left (x \right )}\right ) \]