14.1.9 problem 9

Internal problem ID [2480]
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 : 9
Date solved : Sunday, March 30, 2025 at 12:02:47 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 25

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

\[ y = {\mathrm e}^{-\int _{0}^{t}\sqrt {\textit {\_z1}^{2}+1}\, {\mathrm e}^{-\textit {\_z1}}d \textit {\_z1}} \]

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 32

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

\[ y(t)\to \exp \left (\int _0^t-e^{-K[1]} \sqrt {K[1]^2+1}dK[1]\right ) \]

✓ Sympy. Time used: 3.194 (sec). Leaf size: 32

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 = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (e^{- \int \sqrt {t^{2} + 1} e^{- t}\, dt}\right ) e^{\int \limits ^{0} \sqrt {t^{2} + 1} e^{- t}\, dt} \]