63.22.6 problem 4(f)

Internal problem ID [13236]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 4(f)
Date solved : Tuesday, January 28, 2025 at 05:12:50 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 52

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

✓ Solution by Mathematica

Time used: 0.007 (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*}