74.7.57 problem 64

Internal problem ID [16134]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 64
Date solved : Tuesday, January 28, 2025 at 08:50:37 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} t^{{1}/{3}} y^{{2}/{3}}+t +\left (t^{{2}/{3}} y^{{1}/{3}}+y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 171

dsolve((t^(1/3)*y(t)^(2/3)+t)+(t^(2/3)*y(t)^(1/3)+y(t))*diff(y(t),t)=0,y(t), singsol=all)
 

\[ \int _{\textit {\_b}}^{t}\frac {\textit {\_a}^{{5}/{3}} y^{{4}/{3}}-\textit {\_a}^{{1}/{3}} y^{{8}/{3}}-\textit {\_a}^{3}}{\textit {\_a}^{4}+y^{4}}d \textit {\_a} -\frac {\left (\int _{}^{y}\frac {\left (4 t^{2}+4 t^{{4}/{3}} \textit {\_f}^{{2}/{3}}+4 t^{{2}/{3}} \textit {\_f}^{{4}/{3}}+4 \textit {\_f}^{2}\right ) \left (\int _{\textit {\_b}}^{t}\frac {\textit {\_f}^{{17}/{3}} \textit {\_a}^{{1}/{3}}-2 \textit {\_f}^{{13}/{3}} \textit {\_a}^{{5}/{3}}-2 \textit {\_f}^{{5}/{3}} \textit {\_a}^{{13}/{3}}+\textit {\_f}^{{1}/{3}} \textit {\_a}^{{17}/{3}}+3 \textit {\_a}^{3} \textit {\_f}^{3}}{\left (\textit {\_a}^{4}+\textit {\_f}^{4}\right )^{2}}d \textit {\_a} \right )+3 t^{{2}/{3}} \textit {\_f}^{{1}/{3}}+3 \textit {\_f}}{\textit {\_f}^{2}+t^{{2}/{3}} \textit {\_f}^{{4}/{3}}+t^{{4}/{3}} \textit {\_f}^{{2}/{3}}+t^{2}}d \textit {\_f} \right )}{3}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 1.793 (sec). Leaf size: 53

DSolve[(t^(1/3)*y[t]^(2/3)+t)+(t^(2/3)*y[t]^(1/3)+y[t])*D[y[t],t]==0,y[t],t,IncludeSingularSolutions -> True]
 

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