23.4.248 problem 248

Internal problem ID [6550]
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 : 248
Date solved : Friday, October 03, 2025 at 02:09:29 AM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 2 \left (1-y\right ) y y^{\prime \prime }&=f \left (x \right ) \left (1-y\right ) y y^{\prime }+\left (1-2 y\right ) {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 43

ode:=2*(1-y(x))*y(x)*diff(diff(y(x),x),x) = f(x)*(1-y(x))*y(x)*diff(y(x),x)+(1-2*y(x))*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {4 \,{\mathrm e}^{c_1 \int {\mathrm e}^{\frac {\int f \left (x \right )d x}{2}}d x} c_2^{2}+4 c_2 +{\mathrm e}^{-c_1 \int {\mathrm e}^{\frac {\int f \left (x \right )d x}{2}}d x}}{8 c_2} \]

✓ Mathematica. Time used: 60.056 (sec). Leaf size: 45

ode=2*(1 - y[x])*y[x]*D[y[x],{x,2}] == f[x]*(1 - y[x])*y[x]*D[y[x],x] + (1 - 2*y[x])*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sin ^2\left (\frac {1}{2} \left (\int _1^x-\exp \left (-\int _1^{K[1]}-\frac {1}{2} f(K[1])dK[1]\right ) c_1dK[1]+c_2\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1 - 2*y(x))*Derivative(y(x), x)**2 - (1 - y(x))*f(x)*y(x)*Derivative(y(x), x) + (2 - 2*y(x))*y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(sqrt((y(x) - 1)*(f(x)**2*y(x)**2 - f(x)**2*y(x) + 16*y(x)*Deri