72.1.32 problem 35

Internal problem ID [14553]
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 1.2. page 33
Problem number : 35
Date solved : Thursday, March 13, 2025 at 03:33:54 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.098 (sec). Leaf size: 14

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

\[ y = \tan \left (\frac {t^{2}}{2}+\frac {\pi }{4}\right ) \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 17

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

\[ y(t)\to \tan \left (\frac {1}{4} \left (2 t^2+\pi \right )\right ) \]

✓ Sympy. Time used: 0.299 (sec). Leaf size: 12

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

\[ y{\left (t \right )} = \tan {\left (\frac {t^{2}}{2} + \frac {\pi }{4} \right )} \]