68.1.49 problem 56

Internal problem ID [17116]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 56
Date solved : Thursday, October 02, 2025 at 01:43:24 PM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 18

ode:=diff(y(t),t) = sin(2*t)-cos(2*t); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 25

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

\begin{align*} y(t)&\to \int _0^t(\sin (2 K[1])-\cos (2 K[1]))dK[1] \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = - \frac {\sqrt {2} \sin {\left (2 t + \frac {\pi }{4} \right )}}{2} + \frac {1}{2} \]