74.4.10 problem 10

Internal problem ID [15903]
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 : 10
Date solved : Tuesday, January 28, 2025 at 08:18:08 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

\[ -\frac {1}{t}-\frac {2 \sqrt {y}\, \left (3+y\right )}{9}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 6.480 (sec). Leaf size: 445

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

\begin{align*} y(t)\to \frac {\left (-2+\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}\right ){}^2}{2 \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}} \\ y(t)\to \frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}+\frac {-1-i \sqrt {3}}{\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}}-2 \\ y(t)\to -\frac {1}{4} i \left (\sqrt {3}-i\right ) \sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}+\frac {-1+i \sqrt {3}}{\sqrt [3]{\frac {81}{t^2}+3 \sqrt {\frac {(-3+c_1 t){}^2 \left (\left (16+9 c_1{}^2\right ) t^2-54 c_1 t+81\right )}{t^4}}-\frac {54 c_1}{t}+8+9 c_1{}^2}}-2 \\ \end{align*}