74.6.49 problem 55

Internal problem ID [16072]
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 : 55
Date solved : Tuesday, January 28, 2025 at 08:35:21 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.128 (sec). Leaf size: 134

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

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

✓ Solution by Mathematica

Time used: 39.535 (sec). Leaf size: 177

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

\begin{align*} y(t)\to -\arccos \left (-\frac {c_1 t^2+\sqrt {t^6+t^4-c_1{}^2 t^2}}{t^4+t^2}\right ) \\ y(t)\to \arccos \left (-\frac {c_1 t^2+\sqrt {t^6+t^4-c_1{}^2 t^2}}{t^4+t^2}\right ) \\ y(t)\to -\arccos \left (\frac {\sqrt {t^6+t^4-c_1{}^2 t^2}-c_1 t^2}{t^4+t^2}\right ) \\ y(t)\to \arccos \left (\frac {\sqrt {t^6+t^4-c_1{}^2 t^2}-c_1 t^2}{t^4+t^2}\right ) \\ \end{align*}