68.6.29 problem 30

Internal problem ID [17339]
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 : 30
Date solved : Thursday, October 02, 2025 at 02:05:59 PM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

\begin{align*} 2 t \sin \left (\frac {y}{t}\right )-y \cos \left (\frac {y}{t}\right )+t \cos \left (\frac {y}{t}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 12

ode:=2*t*sin(y(t)/t)-y(t)*cos(y(t)/t)+t*cos(y(t)/t)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \arcsin \left (\frac {c_1}{t^{2}}\right ) t \]

✓ Mathematica. Time used: 11.807 (sec). Leaf size: 21

ode=(2*t*Sin[y[t]/t]-y[t]*Cos[y[t]/t])+t*Cos[y[t]/t]*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t \arcsin \left (\frac {e^{c_1}}{t^2}\right )\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.890 (sec). Leaf size: 22

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

\[ \left [ y{\left (t \right )} = t \left (\pi - \operatorname {asin}{\left (\frac {C_{1}}{t^{2}} \right )}\right ), \ y{\left (t \right )} = t \operatorname {asin}{\left (\frac {C_{1}}{t^{2}} \right )}\right ] \]