67.13.23 problem 20.2 (e)

Internal problem ID [16688]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 20. Euler equations. Additional exercises page 382
Problem number : 20.2 (e)
Date solved : Thursday, October 02, 2025 at 01:37:45 PM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

\begin{align*} y \left (1\right )&=3 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 16

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

\[ y = -3 x \left (-\cos \left (\ln \left (x \right )\right )+\sin \left (\ln \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 17

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+2*y[x]==0; 
ic={y[1]==3,Derivative[1][y][1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 3 x (\cos (\log (x))-\sin (\log (x))) \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = x \left (- 3 \sin {\left (\log {\left (x \right )} \right )} + 3 \cos {\left (\log {\left (x \right )} \right )}\right ) \]