52.9.1 problem 1

Internal problem ID [8379]
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 : 1
Date solved : Monday, January 27, 2025 at 03:49:51 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-5 y\\ y^{\prime }&=4 x \left (t \right )+8 y \end{align*}

✓ Solution by Maple

Time used: 0.317 (sec). Leaf size: 83

dsolve([diff(x(t),t)=3*x(t)-5*y(t),diff(y(t),t)=4*x(t)+8*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_{2} \right ) \\ y &= \frac {{\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_{2} -\cos \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_{1} -5 \sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_{1} -5 \cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_{2} \right )}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 113

DSolve[{D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==4*x[t]+8*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{11} e^{11 t/2} \left (11 c_1 \cos \left (\frac {\sqrt {55} t}{2}\right )-\sqrt {55} (c_1+2 c_2) \sin \left (\frac {\sqrt {55} t}{2}\right )\right ) \\ y(t)\to \frac {1}{55} e^{11 t/2} \left (55 c_2 \cos \left (\frac {\sqrt {55} t}{2}\right )+\sqrt {55} (8 c_1+5 c_2) \sin \left (\frac {\sqrt {55} t}{2}\right )\right ) \\ \end{align*}