74.4.22 problem 22

Internal problem ID [15915]
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 : 22
Date solved : Tuesday, January 28, 2025 at 08:20:29 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {t^{3}}{y \sqrt {\left (1-y^{2}\right ) \left (t^{4}+9\right )}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 50

dsolve(diff(y(t),t)=t^3/(y(t)*sqrt((1-y(t)^2)*(t^4+9))),y(t), singsol=all)
 

\[ -\left (-\frac {y^{2}}{3}+\int _{}^{t}\frac {\textit {\_a}^{3}}{\sqrt {-\left (\textit {\_a}^{4}+9\right ) \left (-1+y^{2}\right )}}d \textit {\_a} +\frac {1}{3}\right ) \sqrt {y+1}\, \sqrt {y-1}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 2.894 (sec). Leaf size: 519

DSolve[D[y[t],t]==t^3/(y[t]*Sqrt[(1-y[t]^2)*(t^4+9)]),y[t],t,IncludeSingularSolutions -> True]
 

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