23.4.165 problem 165

Internal problem ID [6467]
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 : 165
Date solved : Tuesday, September 30, 2025 at 03:01:33 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 53

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

\begin{align*} y &= -a -\sqrt {b \,x^{2}-2 c_1 x +a^{2}+2 c_2} \\ y &= -a +\sqrt {b \,x^{2}-2 c_1 x +a^{2}+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 42.85 (sec). Leaf size: 129

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

\begin{align*} y(x)&\to -\frac {a b+\sqrt {b^3 (x+c_2){}^2-b e^{2 c_1}}}{b}\\ y(x)&\to \frac {-a b+\sqrt {b^3 (x+c_2){}^2-b e^{2 c_1}}}{b}\\ y(x)&\to \frac {-a b+\sqrt {b^3 (x+c_2){}^2}}{b}\\ y(x)&\to -\frac {a b+\sqrt {b^3 (x+c_2){}^2}}{b} \end{align*}

✗ Sympy

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