57.3.1 problem 1

Internal problem ID [14317]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page 19
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:31:27 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 15

ode=D[x[t],t]==t*Cos[t^2]; 
ic={x[0]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} \left (\sin \left (t^2\right )+2\right ) \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 10

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

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