79.4.1 problem (a)

Internal problem ID [21092]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises VII at page 33
Problem number : (a)
Date solved : Thursday, October 02, 2025 at 07:07:26 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

\begin{align*} y^{\prime }-y+y^{2} {\mathrm e}^{x}+5 \,{\mathrm e}^{-x}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=\eta \\ \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 26

ode:=diff(y(x),x)-y(x)+exp(x)*y(x)^2+5*exp(-x) = 0; 
ic:=[y(0) = eta]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (1-2 \tan \left (2 x -\arctan \left (\frac {\eta }{2}-\frac {1}{2}\right )\right )\right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.184 (sec). Leaf size: 55

ode=D[y[x],x]-y[x]+Exp[x]*y[x]^2+5*Exp[-x]==0; 
ic={y[0]==\[Eta]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-x} \left ((-1+2 i) \eta +(-5+(1+2 i) \eta ) e^{4 i x}+5\right )}{-\eta +(\eta -(1-2 i)) e^{4 i x}+(1+2 i)} \end{align*}

✓ Sympy. Time used: 0.678 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(y(x)**2*exp(x) - y(x) + Derivative(y(x), x) + 5*exp(-x),0) 
ics = {y(0): a} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (1 - 2 \tan {\left (2 x - \operatorname {atan}{\left (\frac {a}{2} - \frac {1}{2} \right )} \right )}\right ) e^{- x} \]