47.2.16 problem 16

Internal problem ID [7432]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 16
Date solved : Sunday, March 30, 2025 at 12:03:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Riccati]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 11

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

\[ y = \tan \left (\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 13

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

\[ y(x)\to x \tan (\log (x)+c_1) \]

✓ Sympy. Time used: 0.331 (sec). Leaf size: 27

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

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