23.4.173 problem 173

Internal problem ID [6475]
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 : 173
Date solved : Tuesday, September 30, 2025 at 03:01:43 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 22

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

\[ y = \frac {\left (c_{1}^{2}+a \right ) x^{2}}{4 c_{2}}+c_{1} x +c_{2} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \frac {x^2 \left (a+c_1{}^2\right )}{4 c_2}+c_1 x+c_2\\ y(x)&\to \text {Indeterminate} \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