89.5.15 problem 15

Internal problem ID [24365]
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 43
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:22:14 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 23

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

\[ c_1 +\frac {{\mathrm e}^{v}}{v^{2} x -v-1} = 0 \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 29

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

\[ \text {Solve}\left [x=-\frac {-v(x)-1}{v(x)^2}+\frac {c_1 e^{v(x)}}{v(x)^2},v(x)\right ] \]

✓ Sympy. Time used: 0.656 (sec). Leaf size: 24

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

\[ C_{1} + x v^{2}{\left (x \right )} e^{- v{\left (x \right )}} - \left (v{\left (x \right )} + 1\right ) e^{- v{\left (x \right )}} = 0 \]