23.4.177 problem 177

Internal problem ID [6479]
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 : 177
Date solved : Tuesday, September 30, 2025 at 03:01:46 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y y^{\prime \prime }&=y^{2} \left (a +b y\right )+{y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 76

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

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

✓ Mathematica. Time used: 1.941 (sec). Leaf size: 1375

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

Too large to display

✗ Sympy

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

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