23.6.22 problem 22

Internal problem ID [6821]
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 : 22
Date solved : Tuesday, September 30, 2025 at 03:52:36 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

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

✓ Maple. Time used: 0.258 (sec). Leaf size: 65

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

\begin{align*} y &= -\frac {1}{2} i x^{2}+c_1 x +c_2 \\ y &= \frac {1}{2} i x^{2}+c_1 x +c_2 \\ y &= -\frac {\ln \left (c_1 +x \right ) \left (c_1 +x \right )}{2}+\frac {x^{3}}{12}+\frac {c_1 \,x^{2}}{4}+\frac {\left (1+2 c_2 \right ) x}{2}+\frac {c_1}{2}+c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.336 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to \frac {1}{12} \left (x^3+3 c_1 x^2+3 \left (2+c_1{}^2+4 c_3\right ) x-6 (x+c_1) \log (x+c_1)+12 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.410 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{1} + \frac {C_{2} x^{2}}{4} - \frac {C_{2} \log {\left (C_{2} + x \right )}}{2} + \frac {x^{3}}{12} + x \left (C_{3} - \frac {\log {\left (C_{2} + x \right )}}{2}\right ) \]