76.3.16 problem 16

Internal problem ID [17316]
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 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 16
Date solved : Monday, March 31, 2025 at 03:52:13 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \end{align*}

✓ Maple. Time used: 0.197 (sec). Leaf size: 16

ode:=diff(y(t),t) = 2*t*y(t)^2; 
ic:=y(0) = y__0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {y_{0}}{t^{2} y_{0} -1} \]

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 17

ode=D[y[t],t]==2*t*y[t]^2; 
ic={y[0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {\text {y0}}{1-t^2 \text {y0}} \]

✓ Sympy. Time used: 0.145 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*y(t)**2 + Derivative(y(t), t),0) 
ics = {y(0): y__0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {1}{t^{2} - \frac {1}{y^{0}}} \]