23.4.145 problem 145

Internal problem ID [6447]
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 : 145
Date solved : Tuesday, September 30, 2025 at 02:56:57 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 33

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\frac {{\mathrm e}^{-a x} c_1 a +\left (-b x -c_2 \right ) a +b}{a^{2}}} \\ \end{align*}

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to c_2 e^{-\frac {b x+c_1 e^{-a x}}{a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), x) + b*y(x)**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 -a*y(x)/2 - sqrt((a**2*y(x) + 4*b*y(x) + 4*Derivative(y(x), (x,