14.2.8 problem 8

Internal problem ID [2496]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:03:32 AM
CAS classification : [_separable]

\begin{align*} \sqrt {1+y^{2}}\, y^{\prime }&=\frac {t y^{3}}{\sqrt {t^{2}+1}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.527 (sec). Leaf size: 62

ode:=(1+y(t)^2)^(1/2)*diff(y(t),t) = t*y(t)^3/(t^2+1)^(1/2); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \operatorname {RootOf}\left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_Z}^{2}+1}}\right ) \textit {\_Z}^{2}+\operatorname {arctanh}\left (\frac {\sqrt {2}}{2}\right ) \textit {\_Z}^{2}+\sqrt {2}\, \textit {\_Z}^{2}-2 \sqrt {t^{2}+1}\, \textit {\_Z}^{2}+2 \textit {\_Z}^{2}-\sqrt {\textit {\_Z}^{2}+1}\right ) \]

✗ Mathematica

ode=Sqrt[1+y[t]^2]*D[y[t],t]==t*y[t]^3/Sqrt[1+t^2]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Timed out

✓ Sympy. Time used: 1.573 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*y(t)**3/sqrt(t**2 + 1) + sqrt(y(t)**2 + 1)*Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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