23.4.215 problem 215

Internal problem ID [6517]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 215
Date solved : Tuesday, September 30, 2025 at 03:02:35 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a y^{\prime } \left (-y+x y^{\prime }\right )+x y y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 30

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

\begin{align*} y &= 0 \\ y &= \left (\frac {1}{x^{a +1} c_1 +c_2 a +c_2}\right )^{-\frac {1}{a +1}} \\ \end{align*}

✓ Mathematica. Time used: 2.007 (sec). Leaf size: 22

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

\begin{align*} y(x)&\to c_2 \left (x^{a+1}+c_1\right ){}^{\frac {1}{a+1}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (a*y(x) + sqrt(a*(a*y(x) - 4*x**2*Derivati