68.8.13 problem 13

Internal problem ID [17436]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 13
Date solved : Thursday, October 02, 2025 at 02:21:02 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 45

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

\begin{align*} y &= \frac {1+\sqrt {t^{2} c_1 +1}}{c_1 t} \\ y &= \frac {1-\sqrt {t^{2} c_1 +1}}{c_1 t} \\ \end{align*}

✓ Mathematica. Time used: 1.264 (sec). Leaf size: 80

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

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

✓ Sympy. Time used: 1.230 (sec). Leaf size: 31

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

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