23.6.12 problem 12

Internal problem ID [6811]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 6. THE EQUATION IS NONLINEAR AND OF ORDER GREATER THAN TWO, page 427
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 03:51:54 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+y^{\prime } y^{\prime \prime \prime }&=2 {y^{\prime \prime }}^{2} \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 43

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

\begin{align*} y &= c_1 \\ y &= \ln \left (-\tanh \left (\frac {x}{2}+\frac {c_3}{2}\right )\right ) c_1 -c_2 \\ y &= \ln \left (\tanh \left (\frac {x}{2}+\frac {c_3}{2}\right )\right ) c_1 -c_2 \\ \end{align*}

✓ Mathematica. Time used: 16.023 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to c_2 e^{-x-c_1} \sqrt {e^{2 (x+c_1)}} \arctan \left (e^{x+c_1}\right )+c_3\\ y(x)&\to \frac {\pi c_2}{2}+c_3 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(8*Derivative(y(x), (x, 2))**2 + Derivative(y(x), (x, 3))**