23.4.59 problem 59

Internal problem ID [6361]
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 : 59
Date solved : Friday, October 03, 2025 at 02:05:23 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 45

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

\[ y = \int \left (\left (k -1\right ) \int f \left (x \right ) {\mathrm e}^{-\int g \left (x \right )d x \left (k -1\right )}d x +c_1 \right )^{-\frac {1}{k -1}} {\mathrm e}^{-\int g \left (x \right )d x}d x +c_2 \]

✓ Mathematica. Time used: 34.593 (sec). Leaf size: 84

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

\begin{align*} y(x)&\to \int _1^x\left (\exp \left (-\left ((k-1) \int _1^{K[3]}-g(K[1])dK[1]\right )\right ) \left (c_1-(k-1) \int _1^{K[3]}-\exp \left ((k-1) \int _1^{K[2]}-g(K[1])dK[1]\right ) f(K[2])dK[2]\right )\right ){}^{\frac {1}{1-k}}dK[3]+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(f(x)*Derivative(y(x), x)**k + g(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : multiple generators [_X0, _X0**k] 
No algorithms are implemented to solve equat