66.1.34 problem Problem 48

Internal problem ID [13802]
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 48
Date solved : Wednesday, March 05, 2025 at 10:17:33 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

With initial conditions

\begin{align*} y \left (1\right )&=-1 \end{align*}

✓ Maple. Time used: 0.879 (sec). Leaf size: 61

ode:=y(x) = x*diff(y(x),x)+diff(y(x),x)^2; 
ic:=y(1) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= -\frac {1}{2}+\frac {i \left (x -1\right ) \sqrt {3}}{2}-\frac {x}{2} \\ y &= \frac {\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-2 x +1\right )}{4} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (i \sqrt {3}+2 x -1\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 38

ode=y[x]==x*D[y[x],x]+D[y[x],x]^2; 
ic={y[1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to (-1)^{2/3}-\sqrt [3]{-1} x \\ y(x)\to \sqrt [3]{-1} \left (\sqrt [3]{-1} x-1\right ) \\ \end{align*}

✗ Sympy

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 = {y(1): -1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants