52.9.2 problem 2

Internal problem ID [8380]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number : 2
Date solved : Monday, January 27, 2025 at 03:49:52 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-7 y\\ y^{\prime }&=5 x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 77

dsolve([diff(x(t),t)=4*x(t)-7*y(t),diff(y(t),t)=5*x(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (\cos \left (\sqrt {31}\, t \right ) \sqrt {31}\, c_{1} -\sqrt {31}\, \sin \left (\sqrt {31}\, t \right ) c_{2} +2 \cos \left (\sqrt {31}\, t \right ) c_{2} +2 \sin \left (\sqrt {31}\, t \right ) c_{1} \right )}{5} \\ y &= {\mathrm e}^{2 t} \left (\cos \left (\sqrt {31}\, t \right ) c_{2} +\sin \left (\sqrt {31}\, t \right ) c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 98

DSolve[{D[x[t],t]==4*x[t]-7*y[t],D[y[t],t]==5*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 e^{2 t} \cos \left (\sqrt {31} t\right )+\frac {(2 c_1-7 c_2) e^{2 t} \sin \left (\sqrt {31} t\right )}{\sqrt {31}} \\ y(t)\to c_2 e^{2 t} \cos \left (\sqrt {31} t\right )+\frac {(5 c_1-2 c_2) e^{2 t} \sin \left (\sqrt {31} t\right )}{\sqrt {31}} \\ \end{align*}