14.3.9 problem 11

Internal problem ID [2518]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.9. Exact equations. Excercises page 66
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 02:18:52 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 3 t y+y^{2}+\left (t^{2}+t y\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \end{align*}

✓ Maple. Time used: 1.164 (sec). Leaf size: 21

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

\[ y = \frac {-t^{2}+\sqrt {t^{4}+20}}{t} \]

✓ Mathematica. Time used: 0.74 (sec). Leaf size: 22

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

\[ y(t)\to \frac {\sqrt {t^4+20}}{t}-t \]

✓ Sympy. Time used: 1.332 (sec). Leaf size: 14

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

\[ y{\left (t \right )} = - t + \frac {\sqrt {t^{4} + 20}}{t} \]