8.1.8 problem 8

Internal problem ID [2479]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.2. Linear equations. Excercises page 9
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:36:35 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 24

ode:=diff(y(t),t)+(t^2+1)^(1/2)*y(t) = 0; 
ic:=[y(0) = 5^(1/2)]; 
dsolve([ode,op(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: 0.029 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to \sqrt {5} e^{-\frac {\text {arcsinh}(t)}{2}-\frac {1}{2} \sqrt {t^2+1} t} \end{align*}

✓ Sympy. Time used: 0.214 (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)
 

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