89.2.13 problem 13

Internal problem ID [24278]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 13
Date solved : Thursday, October 02, 2025 at 10:05:54 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 1.505 (sec). Leaf size: 80

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

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

✓ Sympy. Time used: 1.208 (sec). Leaf size: 31

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

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