23.4.148 problem 148

Internal problem ID [6450]
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 : 148
Date solved : Friday, October 03, 2025 at 02:05:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.051 (sec). Leaf size: 37

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

\[ y = c_2 \,{\mathrm e}^{c_1 \int {\mathrm e}^{\int f \left (x \right )d x}d x +\int {\mathrm e}^{\int f \left (x \right )d x} \int {\mathrm e}^{-\int f \left (x \right )d x} g \left (x \right )d x d x} \]

✓ Mathematica. Time used: 0.688 (sec). Leaf size: 61

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

✗ Sympy

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