14.2.6 problem 6

Internal problem ID [2494]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:54:23 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 16

dsolve([t^2*(1+y(t)^2)+2*y(t)*diff(y(t),t)=0,y(0) = 1],y(t), singsol=all)
 

\[ y = \sqrt {2 \,{\mathrm e}^{-\frac {t^{3}}{3}}-1} \]

✓ Solution by Mathematica

Time used: 3.726 (sec). Leaf size: 43

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

\begin{align*} y(t)\to \sqrt {2 e^{-\frac {t^3}{3}}-1} \\ y(t)\to \sqrt {2 e^{-\frac {t^3}{3}}-1} \\ \end{align*}