13.2.9 problem 9

Internal problem ID [2307]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.2. Page 9
Problem number : 9
Date solved : Saturday, March 29, 2025 at 11:53:33 PM
CAS classification : [_separable]

\begin{align*} \sqrt {t^{2}+1}\, y \,{\mathrm e}^{-t}+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 23

ode:=(t^2+1)^(1/2)*y(t)/exp(t)+diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-\int \sqrt {t^{2}+1}\, {\mathrm e}^{-t}d t} \]

✓ Mathematica. Time used: 0.332 (sec). Leaf size: 40

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

\begin{align*} y(t)\to c_1 \exp \left (\int _1^t-e^{-K[1]} \sqrt {K[1]^2+1}dK[1]\right ) \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 3.340 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = C_{1} e^{- \int \sqrt {t^{2} + 1} e^{- t}\, dt} \]