79.6.2 problem (b)

Internal problem ID [21098]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises VIII at page 51
Problem number : (b)
Date solved : Thursday, October 02, 2025 at 07:08:15 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 17

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

\begin{align*} y &= -\frac {x^{2}}{4} \\ y &= c_1 \left (c_1 +x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to c_1 (x+c_1)\\ y(x)&\to -\frac {x^2}{4} \end{align*}

✓ Sympy. Time used: 1.143 (sec). Leaf size: 14

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

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