76.2.21 problem 21

Internal problem ID [17286]
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 : 21
Date solved : Monday, March 31, 2025 at 03:48:49 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-\frac {y}{3}&=3 \cos \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 21

ode:=diff(y(t),t)-1/3*y(t) = 3*cos(t); 
ic:=y(0) = a; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {9 \cos \left (t \right )}{10}+\frac {27 \sin \left (t \right )}{10}+{\mathrm e}^{\frac {t}{3}} \left (a +\frac {9}{10}\right ) \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 35

ode=D[y[t],t]-1/3*y[t]==3*Cos[t]; 
ic={y[0]==a}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{t/3} \left (\int _0^t3 e^{-\frac {K[1]}{3}} \cos (K[1])dK[1]+a\right ) \]

✓ Sympy. Time used: 0.158 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)/3 - 3*cos(t) + Derivative(y(t), t),0) 
ics = {y(0): a} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (a + \frac {9}{10}\right ) e^{\frac {t}{3}} + \frac {27 \sin {\left (t \right )}}{10} - \frac {9 \cos {\left (t \right )}}{10} \]