74.6.13 problem 10 (b)

Internal problem ID [19863]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 10 (b)
Date solved : Thursday, October 02, 2025 at 04:51:34 PM
CAS classification : [_separable]

\begin{align*} 1+v^{2}+\left (u^{2}+1\right ) v v^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

ode:=1+v(u)^2+(u^2+1)*v(u)*diff(v(u),u) = 0; 
dsolve(ode,v(u), singsol=all);
 

\begin{align*} v &= \sqrt {{\mathrm e}^{-2 \arctan \left (u \right )} c_1 -1} \\ v &= -\sqrt {{\mathrm e}^{-2 \arctan \left (u \right )} c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 3.129 (sec). Leaf size: 59

ode=(1+v[u]^2)+(1+u^2)*v[u]*D[v[u],u]==0; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\begin{align*} v(u)&\to -\sqrt {-1+e^{-2 \arctan (u)+2 c_1}}\\ v(u)&\to \sqrt {-1+e^{-2 \arctan (u)+2 c_1}}\\ v(u)&\to -i\\ v(u)&\to i \end{align*}

✓ Sympy. Time used: 0.389 (sec). Leaf size: 32

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq((u**2 + 1)*v(u)*Derivative(v(u), u) + v(u)**2 + 1,0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\[ \left [ v{\left (u \right )} = - \sqrt {C_{1} e^{- 2 \operatorname {atan}{\left (u \right )}} - 1}, \ v{\left (u \right )} = \sqrt {C_{1} e^{- 2 \operatorname {atan}{\left (u \right )}} - 1}\right ] \]