68.4.30 problem 30

Internal problem ID [17210]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 30
Date solved : Thursday, October 02, 2025 at 01:54:51 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {5^{-t}}{y^{2}} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 83

ode:=diff(y(t),t) = 5^(-t)/y(t)^2; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {\left (c_1 \ln \left (5\right )-3 \,5^{-t}\right )^{{1}/{3}}}{\ln \left (5\right )^{{1}/{3}}} \\ y &= -\frac {\left (c_1 \ln \left (5\right )-3 \,5^{-t}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 \ln \left (5\right )^{{1}/{3}}} \\ y &= \frac {\left (c_1 \ln \left (5\right )-3 \,5^{-t}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 \ln \left (5\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 0.647 (sec). Leaf size: 88

ode=D[y[t],t]==5^(-t)/y[t]^2; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sqrt [3]{-\frac {3}{\log (5)}} \sqrt [3]{-5^{-t}+c_1 \log (5)}\\ y(t)&\to \sqrt [3]{-\frac {3\ 5^{-t}}{\log (5)}+3 c_1}\\ y(t)&\to (-1)^{2/3} \sqrt [3]{-\frac {3\ 5^{-t}}{\log (5)}+3 c_1} \end{align*}

✓ Sympy. Time used: 0.695 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), t) - 1/(5**t*y(t)**2),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = \frac {\sqrt [3]{C_{1} - 3 \cdot 5^{- t}}}{\sqrt [3]{\log {\left (5 \right )}}}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - 5^{- t}}}{2 \sqrt [3]{\log {\left (5 \right )}}}, \ y{\left (t \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - 5^{- t}}}{2 \sqrt [3]{\log {\left (5 \right )}}}\right ] \]