23.4.290 problem 293

Internal problem ID [6592]
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 : 293
Date solved : Tuesday, September 30, 2025 at 03:26:40 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} {y^{\prime \prime }}^{2}&=a +b {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.520 (sec). Leaf size: 119

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

\begin{align*} y &= \frac {-\sqrt {-b a}\, x +c_{1} b}{b} \\ y &= \frac {\sqrt {-b a}\, x +c_{1} b}{b} \\ y &= \frac {a \,{\mathrm e}^{-\sqrt {b}\, x}+4 b^{2} c_{2} \left (c_{2} {\mathrm e}^{\sqrt {b}\, x}+c_{1} \right )}{4 b^{2} c_{2}} \\ y &= \frac {4 c_{2}^{2} {\mathrm e}^{-\sqrt {b}\, x} b^{2}+4 c_{1} b^{2} c_{2} +a \,{\mathrm e}^{\sqrt {b}\, x}}{4 b^{2} c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 60.123 (sec). Leaf size: 78

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

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

✗ Sympy

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

Timed Out