68.6.2 problem 2

Internal problem ID [17312]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 2
Date solved : Thursday, October 02, 2025 at 02:02:03 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\begin{align*} y &= \sqrt {-t^{2}+c_1} \\ y &= -\sqrt {-t^{2}+c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 39

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

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

✓ Sympy. Time used: 0.198 (sec). Leaf size: 22

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

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