66.1.18 problem 21

Internal problem ID [15905]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.2. page 33
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:29:39 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{t} y}{1+y^{2}} \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 34

ode:=diff(y(t),t) = exp(t)*y(t)/(1+y(t)^2); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{c_1 +{\mathrm e}^{t}}}{\sqrt {\frac {{\mathrm e}^{2 c_1 +2 \,{\mathrm e}^{t}}}{\operatorname {LambertW}\left ({\mathrm e}^{2 c_1 +2 \,{\mathrm e}^{t}}\right )}}} \]

✓ Mathematica. Time used: 10.815 (sec). Leaf size: 46

ode=D[y[t],t]==Exp[t]*y[t]/(1+y[t]^2); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.430 (sec). Leaf size: 22

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

\[ y{\left (t \right )} = e^{C_{1} + e^{t} - \frac {W\left (e^{2 C_{1} + 2 e^{t}}\right )}{2}} \]