72.1.18 problem 21

Internal problem ID [14618]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.2. page 33
Problem number : 21
Date solved : Tuesday, January 28, 2025 at 06:44:43 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{t} y}{1+y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.140 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{{\mathrm e}^{t}+c_{1}}}{\sqrt {\frac {{\mathrm e}^{2 c_{1} +2 \,{\mathrm e}^{t}}}{\operatorname {LambertW}\left ({\mathrm e}^{2 c_{1} +2 \,{\mathrm e}^{t}}\right )}}} \]

✓ Solution by Mathematica

Time used: 15.303 (sec). Leaf size: 46

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

\begin{align*} y(t)\to -\sqrt {W\left (e^{2 \left (e^t+c_1\right )}\right )} \\ y(t)\to \sqrt {W\left (e^{2 \left (e^t+c_1\right )}\right )} \\ y(t)\to 0 \\ \end{align*}