13.2.8 problem 8

Internal problem ID [2306]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.2. Page 9
Problem number : 8
Date solved : Saturday, March 29, 2025 at 11:53:29 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=\sqrt {5} \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 24

ode:=(t^2+1)^(1/2)*y(t)+diff(y(t),t) = 0; 
ic:=y(0) = 5^(1/2); 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \sqrt {5}\, {\mathrm e}^{-\frac {t \sqrt {t^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (t \right )}{2}} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 34

ode=(t^2+1)^(1/2)*y[t]+D[y[t],t] == 0; 
ic=y[0]==Sqrt[5]; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sqrt {5} e^{-\frac {\text {arcsinh}(t)}{2}-\frac {1}{2} \sqrt {t^2+1} t} \]

✓ Sympy. Time used: 0.366 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(sqrt(t**2 + 1)*y(t) + Derivative(y(t), t),0) 
ics = {y(0): sqrt(5)} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {5} e^{- \frac {t \sqrt {t^{2} + 1}}{2} - \frac {\operatorname {asinh}{\left (t \right )}}{2}} \]