23.3.267 problem 269

Internal problem ID [5981]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 269
Date solved : Tuesday, September 30, 2025 at 02:07:16 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 0.094 (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 = {} 
dsolve(ode,func=y(x),ics=ics)
 

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