76.2.27 problem 27

Internal problem ID [17363]
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 : Tuesday, January 28, 2025 at 10:03:05 AM
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*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 19

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

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

✓ Solution by Mathematica

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

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

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