23.6.15 problem 15

Internal problem ID [6814]
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 : 15
Date solved : Tuesday, September 30, 2025 at 03:52:18 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

\begin{align*} \left (1+{y^{\prime }}^{2}\right ) y^{\prime \prime \prime }&=3 y^{\prime } {y^{\prime \prime }}^{2} \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 67

ode:=(1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x) = 3*diff(y(x),x)*diff(diff(y(x),x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= -\sqrt {-c_2^{2}-2 c_2 x -x^{2}+c_1}+c_3 \\ y &= \sqrt {-c_2^{2}-2 c_2 x -x^{2}+c_1}+c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.96 (sec). Leaf size: 142

ode=(1 + D[y[x],x]^2)*D[y[x],{x,3}] == 3*D[y[x],x]*D[y[x],{x,2}]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3-\frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}\\ y(x)&\to \frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}+c_3\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to c_3-i \sqrt {(x+c_2){}^2}\\ y(x)&\to i \sqrt {(x+c_2){}^2}+c_3 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((Derivative(y(x), x)**2 + 1)*Derivative(y(x), (x, 3)) - 3*Derivative(y(x), x)*Derivative(y(x), (x, 2))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(sqrt(9*Derivative(y(x), (x, 2))**4 - 4*Derivative(y(x), (x, 3)