76.4.26 problem 32

Internal problem ID [17349]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 32
Date solved : Monday, March 31, 2025 at 03:55:39 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 3 x y+y^{2}+\left (x^{2}+x y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 59

ode:=3*x*y(x)+y(x)^2+(x^2+x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-c_1 \,x^{2}-\sqrt {x^{4} c_1^{2}+1}}{c_1 x} \\ y &= \frac {-c_1 \,x^{2}+\sqrt {x^{4} c_1^{2}+1}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 38

ode=(3*x*y[x]+y[x]^2) + (x^2+x*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]+1}{K[1] (K[1]+2)}dK[1]=-2 \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.220 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*y(x) + (x**2 + x*y(x))*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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