72.8.18 problem 31

Internal problem ID [14703]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Review Exercises for chapter 1. page 136
Problem number : 31
Date solved : Thursday, March 13, 2025 at 04:15:36 AM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 23

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

\[ y = -\frac {\cos \left (4 t \right )}{10}+\frac {\sin \left (4 t \right )}{5}+\frac {11 \,{\mathrm e}^{2 t}}{10} \]

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.163 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) - cos(4*t) + Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {11 e^{2 t}}{10} + \frac {\sin {\left (4 t \right )}}{5} - \frac {\cos {\left (4 t \right )}}{10} \]