89.33.30 problem 33

Internal problem ID [25014]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 33
Date solved : Thursday, October 02, 2025 at 11:47:13 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: 0.270 (sec). Leaf size: 107

ode:=(y(x)*diff(diff(y(x),x),x)+1+diff(y(x),x)^2)^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: 22.582 (sec). Leaf size: 194

ode=( y[x]*D[y[x],{x,2}]+1+D[y[x],{x,1}]^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)