76.8.1 problem 1

Internal problem ID [17490]
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 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 10:39:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 55

dsolve([diff(x(t),t)=3*x(t)-2*y(t),diff(y(t),t)=4*x(t)-y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 58

DSolve[{D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==4*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^t (c_1 \cos (2 t)+(c_1-c_2) \sin (2 t)) \\ y(t)\to e^t (c_2 \cos (2 t)+(2 c_1-c_2) \sin (2 t)) \\ \end{align*}