60.7.51 problem 1655 (book 6.64)

Internal problem ID [11601]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1655 (book 6.64)
Date solved : Sunday, March 30, 2025 at 08:31:00 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.199 (sec). Leaf size: 55

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= a \int \sqrt {-\frac {1}{-1+a^{2} \left (x^{2}+2 c_1 \right )^{2}}}\, \left (x^{2}+2 c_1 \right )d x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 60.686 (sec). Leaf size: 308

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

\begin{align*} y(x)\to c_2-\frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}} \\ y(x)\to \frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}+c_2 \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-2**(1/3)*(Derivative(y(x), (x, 2))**2/(a**2*x**2))**(1/3)/4 + 2**(1/3)*sqrt(3)*I*(Derivative(y(x), (x, 2))**2/(a**2*x**2))**(1/3)/4 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method