76.2.27 problem 27

Internal problem ID [17292]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 27
Date solved : Monday, March 31, 2025 at 03:49:40 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 19

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

\[ y = \frac {4 \cos \left (t \right )}{5}+\frac {8 \sin \left (t \right )}{5}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}}}{5} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 35

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

\[ y(t)\to e^{-t/2} \left (\int _0^t2 e^{\frac {K[1]}{2}} \cos (K[1])dK[1]+1\right ) \]

✓ Sympy. Time used: 0.159 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = \frac {8 \sin {\left (t \right )}}{5} + \frac {4 \cos {\left (t \right )}}{5} - \frac {9 e^{- \frac {t}{2}}}{5} \]