72.2.20 problem 16 (v)

Internal problem ID [14576]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number : 16 (v)
Date solved : Thursday, March 13, 2025 at 03:36:25 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

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

\[ y = \frac {1}{-1+c_{1} {\mathrm e}^{-\frac {t^{2}}{2}}} \]

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 46

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (K[1]+1)}dK[1]\&\right ]\left [\frac {t^2}{2}+c_1\right ] \\ y(t)\to -1 \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 2.000 (sec). Leaf size: 66

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

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