36.1.23 problem 23

Internal problem ID [6278]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 23
Date solved : Wednesday, March 05, 2025 at 12:30:45 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{4} \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 10

ode:=diff(y(t),t) = 2*t*cos(y(t))^2; 
ic:=y(0) = 1/4*Pi; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \arctan \left (t^{2}+1\right ) \]

✓ Mathematica. Time used: 0.425 (sec). Leaf size: 11

ode=D[y[t],t]==2*t*Cos[y[t]]^2; 
ic={y[0]==Pi/4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \arctan \left (t^2+1\right ) \]

✓ Sympy. Time used: 1.123 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*cos(y(t))**2 + Derivative(y(t), t),0) 
ics = {y(0): pi/4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 \operatorname {atan}{\left (\frac {\sqrt {t^{4} + 2 t^{2} + 2} - 1}{t^{2} + 1} \right )} \]