23.4.300 problem 303

Internal problem ID [6602]
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 : 303
Date solved : Tuesday, September 30, 2025 at 03:27:24 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.265 (sec). Leaf size: 107

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= 0 \\ y &= -c_1 -\sqrt {-\left (x +c_1 +c_2 \right ) \left (x -c_1 +c_2 \right )} \\ y &= -c_1 +\sqrt {-\left (x +c_1 +c_2 \right ) \left (x -c_1 +c_2 \right )} \\ y &= c_1 -\sqrt {-\left (x +c_1 +c_2 \right ) \left (x -c_1 +c_2 \right )} \\ y &= c_1 +\sqrt {-\left (x +c_1 +c_2 \right ) \left (x -c_1 +c_2 \right )} \\ \end{align*}

✓ Mathematica. Time used: 25.329 (sec). Leaf size: 194

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

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

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(sqrt(3)*(3*sqrt(3)*sqrt(27*y(x)**4*Derivative(y(x), (x, 2)