76.6.8 problem 8

Internal problem ID [17463]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 10:39:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 45

dsolve([diff(x(t),t)=3*x(t)-4*y(t),diff(y(t),t)=x(t)+3*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.069 (sec). Leaf size: 157

DSolve[{D[x[t],t]==3*x[t]-4*y[t]+4,D[y[t],t]==x[t]+3*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{3 t} \left (-2 \sin (2 t) \int _1^t-2 e^{-3 K[2]} \sin (2 K[2])dK[2]+\cos (2 t) \int _1^t4 e^{-3 K[1]} \cos (2 K[1])dK[1]+c_1 \cos (2 t)-2 c_2 \sin (2 t)\right ) \\ y(t)\to \frac {1}{2} e^{3 t} \left (2 \cos (2 t) \int _1^t-2 e^{-3 K[2]} \sin (2 K[2])dK[2]+\sin (2 t) \int _1^t4 e^{-3 K[1]} \cos (2 K[1])dK[1]+2 c_2 \cos (2 t)+c_1 \sin (2 t)\right ) \\ \end{align*}