74.1.56 problem 77

Internal problem ID [15836]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 77
Date solved : Tuesday, January 28, 2025 at 08:08:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 75

dsolve([diff(x(t),t)=4*y(t),diff(y(t),t)=-x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_{2} -\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_{1} +\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 93

DSolve[{D[x[t],t]==4*y[t],D[y[t],t]==-x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-t} \left (3 c_1 \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1+4 c_2) \sin \left (\sqrt {3} t\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t} \left (3 c_2 \cos \left (\sqrt {3} t\right )-\sqrt {3} (c_1+c_2) \sin \left (\sqrt {3} t\right )\right ) \\ \end{align*}