23.4.155 problem 155

Internal problem ID [6457]
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 : 155
Date solved : Tuesday, September 30, 2025 at 02:57:28 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y y^{\prime \prime }&=b y^{3}+a {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 89

ode:=y(x)*diff(diff(y(x),x),x) = b*y(x)^3+a*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ \left (2 a -3\right ) \int _{}^{y}\frac {1}{\sqrt {\left (2 a -3\right ) \left (-2 b \,\textit {\_a}^{3}+c_1 \,\textit {\_a}^{2 a}\right )}}d \textit {\_a} -x -c_2 &= 0 \\ \left (-2 a +3\right ) \int _{}^{y}\frac {1}{\sqrt {\left (2 a -3\right ) \left (-2 b \,\textit {\_a}^{3}+c_1 \,\textit {\_a}^{2 a}\right )}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 2.161 (sec). Leaf size: 802

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

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