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 : Sunday, March 30, 2025 at 12:03:25 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*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 16

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

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

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

ode=t^2*(1+y[t]^2)+2*y[t]*D[y[t],t]==0; 
ic={y[0]==1}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.782 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*(y(t)**2 + 1) + 2*y(t)*Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {-1 + 2 e^{- \frac {t^{3}}{3}}} \]