78.1.9 problem 1.i

Internal problem ID [20935]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 1.i
Date solved : Thursday, October 02, 2025 at 06:49:34 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 19

ode:=diff(y(x),x)+y(x) = sin(x); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 21

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

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 19

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

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