66.1.37 problem Problem 51

Internal problem ID [13805]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 51
Date solved : Wednesday, March 05, 2025 at 10:17:39 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 79

ode:=y(x) = x^2+2*x*diff(y(x),x)+1/2*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

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

Timed out

✓ Sympy. Time used: 1.823 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - x^{2} + \frac {\left (C_{1} - 2 x\right )^{2}}{8} \]