62.33.2 problem Ex 2

Internal problem ID [12918]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 57. Dependent variable absent. Page 132
Problem number : Ex 2
Date solved : Monday, March 31, 2025 at 07:24:33 AM
CAS classification : [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 94

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

\begin{align*} y &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y &= -\frac {\sqrt {-x^{2}+1}\, x^{2}}{6}-\frac {\sqrt {-x^{2}+1}}{3}-\frac {x \arcsin \left (x \right )}{2}+c_{1} x +c_{2} \\ y &= \frac {\sqrt {c_{1}^{2}-1}\, x^{3}}{6}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.164 (sec). Leaf size: 75

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

\begin{align*} y(x)\to \frac {c_1 x^3}{6}-\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ y(x)\to \frac {c_1 x^3}{6}+\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ \end{align*}

✗ Sympy

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

Timed Out