82.3.2 problem Ex. 2

Internal problem ID [18661]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 16
Problem number : Ex. 2
Date solved : Monday, March 31, 2025 at 05:50:00 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 2.369 (sec). Leaf size: 80

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

\begin{align*} y(x)\to \frac {e^{2 c_1}-\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}}{x} \\ y(x)\to \frac {\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}+e^{2 c_1}}{x} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.874 (sec). Leaf size: 31

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

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