23.4.291 problem 294

Internal problem ID [6593]
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 : 294
Date solved : Tuesday, September 30, 2025 at 03:26:44 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime }-x y^{\prime \prime }+{y^{\prime \prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 28

ode:=diff(y(x),x)-x*diff(diff(y(x),x),x)+diff(diff(y(x),x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3}}{12}+c_{1} \\ y &= \frac {1}{2} c_{1} x^{2}-c_{1}^{2} x +c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 24

ode=D[y[x],x] - x*D[y[x],{x,2}] + D[y[x],{x,2}]^2 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 x^2}{2}-c_1{}^2 x+c_2 \end{align*}

✓ Sympy. Time used: 1.278 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} - \frac {C_{2}^{2} x}{4} - \frac {C_{2} x^{2}}{4} \]