54.7.53 problem 1657 (book 6.66)

Internal problem ID [12902]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1657 (book 6.66)
Date solved : Friday, October 03, 2025 at 03:50:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.795 (sec). Leaf size: 795

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 42.196 (sec). Leaf size: 9706

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

Too large to display

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-(-b**2*x**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*b*c*x/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*b*x*y(x)/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - c**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*c*y(x)/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) + 1 + Derivative(y(x), (x, 2))**2/(4*a**2*b**2*x**2 + 8*a**2*b*c*x + 8*a**2*b*x*y(x) + 4*a**2*c**2 + 8*a**2*c*y(x) + 4*a**2*y(x)**2) - y(x)**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2))**(1/3)/2 + sqrt(3)*I*(-b**2*x**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*b*c*x/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*b*x*y(x)/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - c**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) - 2*c*y(x)/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2) + 1 + Derivative(y(x), (x, 2))**2/(4*a**2*b**2*x**2 + 8*a**2*b*c*x + 8*a**2*b*x*y(x) + 4*a**2*c**2 + 8*a**2*c*y(x) + 4*a**2*y(x)**2) - y(x)**2/(b**2*x**2 + 2*b*c*x + 2*b*x*y(x) + c**2 + 2*c*y(x) + y(x)**2))**(1/3)/2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method