23.4.170 problem 170

Internal problem ID [6472]
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 : 170
Date solved : Friday, October 03, 2025 at 02:09:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 82

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

\[ \text {Solve}\left [\left \{x=\int \frac {\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{f(K[3])}dK[3]-c_1\right )}{f(K[4])} \, dK[4]+c_2,y(x)=x-\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{f(K[3])}dK[3]-c_1\right )\right \},\{y(x),K[4]\}\right ] \]

✗ Sympy

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

NotImplementedError : 
No algorithms are implemented to solve equation _Dummy_37*x - _Dummy_37*y(x)