23.4.127 problem 127

Internal problem ID [6429]
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 : 127
Date solved : Tuesday, September 30, 2025 at 02:56:36 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 66

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

\begin{align*} y &= \frac {c_1 \left (-{\mathrm e}^{\frac {c_2 +x}{c_1}} a^{2}+{\mathrm e}^{\frac {-c_2 -x}{c_1}}\right )}{2} \\ y &= -\frac {c_1 \left ({\mathrm e}^{\frac {-c_2 -x}{c_1}} a^{2}-{\mathrm e}^{\frac {c_2 +x}{c_1}}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 60.111 (sec). Leaf size: 79

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

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

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(a**2 + y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x