66.1.11 problem Problem 11

Internal problem ID [13779]
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 11
Date solved : Wednesday, March 05, 2025 at 10:15:55 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 55

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

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

✓ Sympy. Time used: 0.520 (sec). Leaf size: 27

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

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