59.1.8 problem 5.4 (iii)

Internal problem ID [14992]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 5, Trivial differential equations. Exercises page 33
Problem number : 5.4 (iii)
Date solved : Thursday, October 02, 2025 at 09:58:09 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 13

ode:=diff(x(t),t) = 2*sin(t)^2; 
ic:=[x(1/4*Pi) = 1/4*Pi]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = t +\frac {1}{2}-\frac {\sin \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 16

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

\begin{align*} x(t)&\to t-\sin (t) \cos (t)+\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = t - \sin {\left (t \right )} \cos {\left (t \right )} + \frac {1}{2} \]