74.7.16 problem 16

Internal problem ID [16014]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 16
Date solved : Thursday, March 13, 2025 at 07:13:49 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 45

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

\begin{align*} y &= \sqrt {-t \sqrt {t^{2}+1}-\operatorname {arcsinh}\left (t \right )+c_{1}} \\ y &= -\sqrt {-t \sqrt {t^{2}+1}-\operatorname {arcsinh}\left (t \right )+c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 1.799 (sec). Leaf size: 61

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

\begin{align*} y(t)\to -\sqrt {-\text {arcsinh}(t)-\sqrt {t^2+1} t+2 c_1} \\ y(t)\to \sqrt {-\text {arcsinh}(t)-\sqrt {t^2+1} t+2 c_1} \\ \end{align*}

✓ Sympy. Time used: 0.478 (sec). Leaf size: 42

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

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