23.4.212 problem 212

Internal problem ID [6514]
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 : 212
Date solved : Tuesday, September 30, 2025 at 03:02:33 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 164

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

\begin{align*} y &= 0 \\ y &= \frac {3^{{1}/{3}} {\left (\left (-x^{2 a +1} c_1 +x^{3 a} c_2 \left (a -1\right )\right ) \left (a -1\right )^{2}\right )}^{{1}/{3}} x^{-a}}{a -1} \\ y &= -\frac {{\left (\left (-x^{2 a +1} c_1 +x^{3 a} c_2 \left (a -1\right )\right ) \left (a -1\right )^{2}\right )}^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x^{-a}}{2 a -2} \\ y &= \frac {{\left (\left (-x^{2 a +1} c_1 +x^{3 a} c_2 \left (a -1\right )\right ) \left (a -1\right )^{2}\right )}^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{-a}}{2 a -2} \\ \end{align*}

✓ Mathematica. Time used: 3.549 (sec). Leaf size: 29

ode=a*y[x]*D[y[x],x] + 2*x*D[y[x],x]^2 + 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 \sqrt [3]{3 x^{1-a}-a c_1+c_1} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), x) + x*y(x)*Derivative(y(x), (x, 2)) + 2*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**2*y(x) - 8*x**2*Deriva