76.5.16 problem 16

Internal problem ID [17365]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 16
Date solved : Monday, March 31, 2025 at 04:07:50 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} t^{2} y^{\prime }+2 t y-y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 60

ode:=t^2*diff(y(t),t)+2*t*y(t)-y(t)^3 = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {5}\, \sqrt {5 t^{6} c_1 +2 t}}{5 c_1 \,t^{5}+2} \\ y &= -\frac {\sqrt {5}\, \sqrt {5 t^{6} c_1 +2 t}}{5 c_1 \,t^{5}+2} \\ \end{align*}

✓ Mathematica. Time used: 0.561 (sec). Leaf size: 63

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

\begin{align*} y(t)\to -\frac {\sqrt {5} \sqrt {t}}{\sqrt {2+5 c_1 t^5}} \\ y(t)\to \frac {\sqrt {5} \sqrt {t}}{\sqrt {2+5 c_1 t^5}} \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.576 (sec). Leaf size: 39

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

\[ \left [ y{\left (t \right )} = - \sqrt {5} \sqrt {\frac {t}{C_{1} t^{5} + 2}}, \ y{\left (t \right )} = \sqrt {5} \sqrt {\frac {t}{C_{1} t^{5} + 2}}\right ] \]