23.4.56 problem 56

Internal problem ID [6358]
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 : 56
Date solved : Tuesday, September 30, 2025 at 02:52:28 PM
CAS classification : [[_2nd_order, _with_exponential_symmetries], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (a x +b y\right ) {y^{\prime }}^{3}+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 35

ode:=(a*x+b*y(x))*diff(y(x),x)^3+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (-2 \,{\mathrm e}^{\textit {\_Z}} a^{{3}/{2}} x -{\mathrm e}^{2 \textit {\_Z}} c_2 a -2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} b +c_1 a \right )}{\sqrt {a}} \]

✓ Mathematica. Time used: 1.189 (sec). Leaf size: 158

ode=(a*x + b*y[x])*D[y[x],x]^3 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [-\frac {b y(x)}{a}-\frac {b \log \left (a^2 x+\sqrt {a} \sqrt {a^3 x^2+2 a^2 b x y(x)+b^2 \left (a y(x)^2-2 a c_1+1\right )}+a b y(x)\right )}{a^{3/2}}=c_2,y(x)\right ]\\ \text {Solve}\left [-\frac {b y(x)}{a}+\frac {b \log \left (a^2 x+\sqrt {a} \sqrt {a^3 x^2+2 a^2 b x y(x)+b^2 \left (a y(x)^2-2 a c_1+1\right )}+a b y(x)\right )}{a^{3/2}}=c_2,y(x)\right ] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (-Derivative(y(x), (x, 2))/(a*x + b*y(x)))**(1/3)/2 - sqrt(3)*I*