76.2.29 problem 29

Internal problem ID [17294]
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 : 29
Date solved : Monday, March 31, 2025 at 03:49:45 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+\frac {y}{4}&=3+2 \cos \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 24

ode:=diff(y(t),t)+1/4*y(t) = 3+2*cos(2*t); 
ic:=y(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = 12+\frac {8 \cos \left (2 t \right )}{65}+\frac {64 \sin \left (2 t \right )}{65}-\frac {788 \,{\mathrm e}^{-\frac {t}{4}}}{65} \]

✓ Mathematica. Time used: 0.163 (sec). Leaf size: 38

ode=D[y[t],t]+1/4*y[t]==3+2*Cos[2*t]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-t/4} \int _0^te^{\frac {K[1]}{4}} (2 \cos (2 K[1])+3)dK[1] \]

✓ Sympy. Time used: 0.547 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t)/4 - 2*cos(2*t) + Derivative(y(t), t) - 3,0) 
ics = {y(0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {64 \sin {\left (2 t \right )}}{65} + \frac {8 \cos {\left (2 t \right )}}{65} + 12 - \frac {788 e^{- \frac {t}{4}}}{65} \]