23.4.48 problem 48

Internal problem ID [6350]
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 : 48
Date solved : Tuesday, September 30, 2025 at 02:52:17 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} f \left (y\right ) y^{\prime }+g \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 43

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

\begin{align*} y &= c_1 \\ -\int _{}^{y}\frac {{\mathrm e}^{\int g \left (\textit {\_g} \right )d \textit {\_g}}}{\int f \left (\textit {\_g} \right ) {\mathrm e}^{\int g \left (\textit {\_g} \right )d \textit {\_g}}d \textit {\_g} -c_1}d \textit {\_g} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 258

ode=f[y[x]]*D[y[x],x] + g[y[x]]*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 {$\#$1}}\frac {1}{\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) c_1+\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) \int _1^{K[3]}-\exp \left (-\int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) (-c_1)+\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) \int _1^{K[3]}-\exp \left (-\int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) c_1+\exp \left (\int _1^{K[3]}-g(K[1])dK[1]\right ) \int _1^{K[3]}-\exp \left (-\int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]}dK[3]\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(f(y(x))*Derivative(y(x), x) + g(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(f(y(x))**2 - 4*g(y(x))*Derivative(y(x), (x, 2))) - f(y(x)