67.1.31 problem 2.5 (b i)

Internal problem ID [16296]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number : 2.5 (b i)
Date solved : Thursday, October 02, 2025 at 10:45:28 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 19

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

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

✓ Sympy. Time used: 0.062 (sec). Leaf size: 10

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

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