23.4.294 problem 297

Internal problem ID [6596]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 297
Date solved : Tuesday, September 30, 2025 at 03:27:19 PM
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: 0.073 (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: 0.104 (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