78.2.23 problem 8.b

Internal problem ID [20975]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 8.b
Date solved : Thursday, October 02, 2025 at 07:00:53 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+3 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: 0.030 (sec). Leaf size: 13

ode:=x^2*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+3*y(x) = 0; 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {1}{2} x^{3}+\frac {3}{2} x \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 15

ode=x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+3*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 -\frac {1}{2} x \left (x^2-3\right ) \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 12

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

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