74.8.5 problem 5

Internal problem ID [16141]
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 : 5
Date solved : Tuesday, January 28, 2025 at 08:51:50 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{5 t}}{y^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 136

dsolve(diff(y(t),t)=exp(5*t)/y(t)^4,y(t), singsol=all)
 

\begin{align*} y &= \left ({\mathrm e}^{5 t}+c_{1} \right )^{{1}/{5}} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left ({\mathrm e}^{5 t}+c_{1} \right )^{{1}/{5}}}{4} \\ y &= -\frac {\left (-i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left ({\mathrm e}^{5 t}+c_{1} \right )^{{1}/{5}}}{4} \\ y &= \frac {\left (-i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left ({\mathrm e}^{5 t}+c_{1} \right )^{{1}/{5}}}{4} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left ({\mathrm e}^{5 t}+c_{1} \right )^{{1}/{5}}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.754 (sec). Leaf size: 117

DSolve[D[y[t],t]==Exp[5*t]/y[t]^4,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \sqrt [5]{e^{5 t}+5 c_1} \\ y(t)\to -\sqrt [5]{-1} \sqrt [5]{e^{5 t}+5 c_1} \\ y(t)\to (-1)^{2/5} \sqrt [5]{e^{5 t}+5 c_1} \\ y(t)\to -(-1)^{3/5} \sqrt [5]{e^{5 t}+5 c_1} \\ y(t)\to (-1)^{4/5} \sqrt [5]{e^{5 t}+5 c_1} \\ \end{align*}