74.6.15 problem 16

Internal problem ID [16038]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 08:28:36 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 16

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

\[ -y^{4}-y c_{1} +t = 0 \]

✓ Solution by Mathematica

Time used: 60.146 (sec). Leaf size: 1073

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

\begin{align*} y(t)\to \frac {\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}{2 \sqrt [3]{6}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{\frac {2}{3}} t}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}{\sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{6} c_1}{\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}} \\ y(t)\to \frac {\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}{2 \sqrt [3]{6}}+\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{\frac {2}{3}} t}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}{\sqrt [3]{2} 3^{2/3}}-\frac {2 \sqrt [3]{6} c_1}{\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}} \\ y(t)\to -\frac {\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}{2 \sqrt [3]{6}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{\frac {2}{3}} t}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}{\sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{6} c_1}{\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}} \\ y(t)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{\frac {2}{3}} t}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}{\sqrt [3]{2} 3^{2/3}}+\frac {2 \sqrt [3]{6} c_1}{\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}}-\frac {\sqrt {\frac {-8 \sqrt [3]{3} t+\sqrt [3]{2} \left (9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}\right ){}^{2/3}}{\sqrt [3]{9 c_1{}^2-\sqrt {768 t^3+81 c_1{}^4}}}}}{2 \sqrt [3]{6}} \\ \end{align*}