44.1.26 problem 3(b)

Internal problem ID [9084]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS. Page 9
Problem number : 3(b)
Date solved : Tuesday, September 30, 2025 at 06:03:51 PM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 12

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

\[ y = -\frac {\cos \left (2 x \right )}{2}+\frac {3}{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {3}{2} - \frac {\cos {\left (2 x \right )}}{2} \]