75.2.2 problem 2

Internal problem ID [19896]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 2
Date solved : Thursday, October 02, 2025 at 05:00:28 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }-{y^{\prime }}^{2}-y {y^{\prime }}^{3}&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)-diff(y(x),x)^2-y(x)*diff(y(x),x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \\ y-{\mathrm e}^{-y} c_1 -\frac {y^{2}}{2}-x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 96

ode=D[y[x],{x,2}]-D[y[x],x]^2-y[x]*D[y[x],x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*Derivative(y(x), x)**3 - Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (sqrt((-27*Derivative(y(x), (x, 2))/y(x) + 2/y(x)**3)**2 - 4/y(x)**6)/2 - 27*Derivative(y(x), (x, 2))/(2*y(x)) + y(x)**(-3))**(1/3)/3 + Derivative(y(x), x) + 1/(3*y(x)) + 1/(3*(sqrt((-27*Derivative(y(x), (x, 2))/y(x) + 2/y(x)**3)**2 - 4/y(x)**6)/2 - 27*Derivative(y(x), (x, 2))/(2*y(x)) + y(x)**(-3))**(1/3)*y(x)**2) cannot be solved by the factorable group method