60.7.208 problem 1834 (book 6.243)

Internal problem ID [11758]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1834 (book 6.243)
Date solved : Wednesday, March 05, 2025 at 03:05:52 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.197 (sec). Leaf size: 124

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

\begin{align*} y &= \frac {\operatorname {csgn}\left (\sec \left (\frac {-x +c_{1}}{b}\right )\right ) \sin \left (\frac {-x +c_{1}}{b}\right ) \operatorname {csgn}\left (a \right ) b}{a} \\ y &= -\frac {\operatorname {csgn}\left (\sec \left (\frac {-x +c_{1}}{b}\right )\right ) \sin \left (\frac {-x +c_{1}}{b}\right ) \operatorname {csgn}\left (a \right ) b}{a} \\ y &= -\frac {b}{a} \\ y &= \frac {b}{a} \\ y &= c_{1} \\ y &= \frac {b \left ({\mathrm e}^{\frac {\sqrt {a^{2} c_{1}^{2}-1}\, \left (x +c_{2} \right )}{b}}-c_{1} \right )}{\sqrt {a^{2} c_{1}^{2}-1}} \\ \end{align*}

✓ Mathematica. Time used: 82.17 (sec). Leaf size: 81

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

\begin{align*} y(x)\to \frac {b \left (e^{\frac {\sqrt {-1+a^2 c_1{}^2} (x+c_2)}{b}}-c_1\right )}{\sqrt {-1+a^2 c_1{}^2}} \\ y(x)\to c_1 e^{c_2 x}-\frac {\sqrt {b^2+\frac {1}{c_2{}^2}}}{a} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt(2*y(x)*Derivative(y(x), (x, 2)) + sqrt(4*a**2*b**2*Derivative(y(x), (x, 2))**2 + 4*a**2*y(x)*Derivative(y(x), (x, 2)) + 1)/a**2 + a**(-2))/2 + Derivative(y(x), x) cannot be solved by the factorable group method