56.33.5 problem Ex 5

Internal problem ID [14273]
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 5
Date solved : Thursday, October 02, 2025 at 09:27:32 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.088 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 58

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

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

✗ Sympy

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

Timed Out