23.4.67 problem 67

Internal problem ID [6369]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 67
Date solved : Friday, October 03, 2025 at 02:05:35 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 }&=a \left (b +c x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.598 (sec). Leaf size: 797

ode:=diff(diff(y(x),x),x) = a*(b+c*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: 43.471 (sec). Leaf size: 9706

ode=D[y[x],{x,2}] == a*(b + c*x + y[x])*(1 + D[y[x],x]^2)^(3/2); 
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(-a*(Derivative(y(x), x)**2 + 1)**(3/2)*(b + c*x + y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-(-b**2/(b**2 + 2*b*c*x + 2*b*y(x) + c**2*x**2 + 2*c*x*y(x