60.8.11 problem 1847

Internal problem ID [11772]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1847
Date solved : Sunday, March 30, 2025 at 09:14:44 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

\begin{align*} \left ({y^{\prime }}^{2}+1\right ) y^{\prime \prime \prime }-3 y^{\prime } {y^{\prime \prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 65

ode:=(1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x)-3*diff(y(x),x)*diff(diff(y(x),x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= -\sqrt {-c_2^{2}-2 c_2 x -x^{2}+c_1}+c_3 \\ y &= \sqrt {-c_2^{2}-2 c_2 x -x^{2}+c_1}+c_3 \\ \end{align*}

✓ Mathematica. Time used: 1.133 (sec). Leaf size: 142

ode=-3*D[y[x],x]*D[y[x],{x,2}]^2 + (1 + D[y[x],x]^2)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : The given ODE -(sqrt(9*Derivative(y(x), (x, 2))**4 - 4*Derivative(y(x), (x, 3))**2) + 3*Derivative(y(x), (x, 2))**2)/(2*Derivative(y(x), (x, 3))) + Derivative(y(x), x) cannot be solved by the factorable group method