74.8.25 problem 25

Internal problem ID [16090]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 25
Date solved : Monday, March 31, 2025 at 02:41:32 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 72

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

\begin{align*} y &= \frac {\left (8 \,{\mathrm e}^{-3 t} c_1 +6 \,{\mathrm e}^{t}\right )^{{1}/{3}}}{2} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (8 \,{\mathrm e}^{-3 t} c_1 +6 \,{\mathrm e}^{t}\right )^{{1}/{3}}}{4} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (8 \,{\mathrm e}^{-3 t} c_1 +6 \,{\mathrm e}^{t}\right )^{{1}/{3}}}{4} \\ \end{align*}

✓ Mathematica. Time used: 8.253 (sec). Leaf size: 96

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

\begin{align*} y(t)\to \frac {\sqrt [3]{3 e^t+4 c_1 e^{-3 t}}}{2^{2/3}} \\ y(t)\to -\frac {\sqrt [3]{-1} \sqrt [3]{3 e^t+4 c_1 e^{-3 t}}}{2^{2/3}} \\ y(t)\to \left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{3 e^t+4 c_1 e^{-3 t}} \\ \end{align*}

✓ Sympy. Time used: 1.170 (sec). Leaf size: 90

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

\[ \left [ y{\left (t \right )} = \frac {\sqrt [3]{2} \sqrt [3]{C_{1} e^{- 3 t} + 3 e^{t}}}{2}, \ y{\left (t \right )} = \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} e^{- 3 t} + 3 e^{t}}}{4}, \ y{\left (t \right )} = \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} e^{- 3 t} + 3 e^{t}}}{4}\right ] \]