78.1.2 problem 1.b

Internal problem ID [20928]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 1.b
Date solved : Thursday, October 02, 2025 at 06:49:18 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 16

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

\[ y = -\frac {3}{t^{3}+3 t -3} \]

✓ Mathematica. Time used: 0.115 (sec). Leaf size: 17

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

\begin{align*} y(t)&\to -\frac {3}{t^3+3 t-3} \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 14

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

\[ y{\left (t \right )} = - \frac {3}{t^{3} + 3 t - 3} \]