23.4.69 problem 69

Internal problem ID [6371]
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 : 69
Date solved : Tuesday, September 30, 2025 at 02:53:34 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x) = f(diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \int \operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_f} \right )}d \textit {\_f} +c_1 \right )d x +c_2 \]

✓ Mathematica. Time used: 0.825 (sec). Leaf size: 35

ode=D[y[x],{x,2}] == f[D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{f(K[1])}dK[1]\&\right ][c_1+K[2]]dK[2]+c_2 \end{align*}

✗ Sympy

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

NotImplementedError : 
No algorithms are implemented to solve equation _Dummy_37 - f(_X0)