68.5.53 problem 60 (a)

Internal problem ID [17304]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.3, page 49
Problem number : 60 (a)
Date solved : Thursday, October 02, 2025 at 02:01:16 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 21

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

\[ y = -\frac {4 \cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5}+{\mathrm e}^{-\frac {t}{2}} \left (a +\frac {4}{5}\right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to e^{-t/2} \left (\int _0^te^{\frac {K[1]}{2}} \sin (K[1])dK[1]+a\right ) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \left (a + \frac {4}{5}\right ) e^{- \frac {t}{2}} + \frac {2 \sin {\left (t \right )}}{5} - \frac {4 \cos {\left (t \right )}}{5} \]