66.2.21 problem Problem 30

Internal problem ID [13849]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 30
Date solved : Monday, March 31, 2025 at 08:15:08 AM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=\frac {y y^{\prime }}{\sqrt {x^{2}+1}} \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 63

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

\begin{align*} y &= 0 \\ y &= \sqrt {c_1 x \sqrt {x^{2}+1}+c_1 \,x^{2}+c_1 \,\operatorname {arcsinh}\left (x \right )+2 c_2} \\ y &= -\sqrt {c_1 x \sqrt {x^{2}+1}+c_1 \,x^{2}+c_1 \,\operatorname {arcsinh}\left (x \right )+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 66.228 (sec). Leaf size: 47

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

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {e^{\text {arcsinh}(K[1])}}{\text {arcsinh}(K[1])+c_1+K[1] \left (K[1]+\sqrt {K[1]^2+1}\right )}dK[1]\right ) \]

✗ Sympy

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

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