23.4.131 problem 131

Internal problem ID [6433]
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 : 131
Date solved : Tuesday, September 30, 2025 at 02:56:46 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y y^{\prime \prime }&=\operatorname {a0} +\operatorname {a1} y+y^{3} \left (\operatorname {a2} +\operatorname {a3} y\right )+{y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 83

ode:=y(x)*diff(diff(y(x),x),x) = a0+a1*y(x)+y(x)^3*(a2+a3*y(x))+diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4} \operatorname {a3} +2 \textit {\_a}^{3} \operatorname {a2} +c_1 \,\textit {\_a}^{2}-2 \operatorname {a1} \textit {\_a} -\operatorname {a0}}}d \textit {\_a} -x -c_2 &= 0 \\ -\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4} \operatorname {a3} +2 \textit {\_a}^{3} \operatorname {a2} +c_1 \,\textit {\_a}^{2}-2 \operatorname {a1} \textit {\_a} -\operatorname {a0}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 3.65 (sec). Leaf size: 2633

ode=y[x]*D[y[x],{x,2}] == a0 + a1*y[x] + y[x]^3*(a2 + a3*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") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(-a0 - a1*y(x) - (a2 + a3*y(x))*y(x)**3 + 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(-a0 - a1*y(x) - a2*y(x)**3 - a3*y(x)**4 + y(x)*Derivative(