23.4.40 problem 40

Internal problem ID [6342]
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 : 40
Date solved : Tuesday, September 30, 2025 at 02:51:57 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} b y+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 79

ode:=b*y(x)+a*diff(y(x),x)^2+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -2 a \int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 b \textit {\_a} a +2 b}}d \textit {\_a} -x -c_2 &= 0 \\ 2 a \int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 b \textit {\_a} a +2 b}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.658 (sec). Leaf size: 314

ode=b*y[x] + a*D[y[x],x]^2 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[1]} c_1 a^2-2 b K[1] a+b}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[2]} c_1 a^2-2 b K[2] a+b}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[1]} (-c_1) a^2-2 b K[1] a+b}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[1]} c_1 a^2-2 b K[1] a+b}}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[2]} (-c_1) a^2-2 b K[2] a+b}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[2]} c_1 a^2-2 b K[2] a+b}}dK[2]\&\right ][x+c_2] \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) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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