74.4.30 problem 30

Internal problem ID [15923]
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 : Tuesday, January 28, 2025 at 08:22:50 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 83

dsolve(diff(y(t),t)=5^(-t)/y(t)^2,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*}

✓ Solution by Mathematica

Time used: 0.915 (sec). Leaf size: 88

DSolve[D[y[t],t]==5^(-t)/y[t]^2,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*}