52.10.33 problem 36

Internal problem ID [8427]
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 8.2. Page 346
Problem number : 36
Date solved : Monday, January 27, 2025 at 04:00:23 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+5 y\\ y^{\prime }&=-2 x \left (t \right )+6 y \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 58

dsolve([diff(x(t),t)=4*x(t)+5*y(t),diff(y(t),t)=-2*x(t)+6*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} \left (\sin \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \right ) \\ y &= \frac {{\mathrm e}^{5 t} \left (\sin \left (3 t \right ) c_{1} -3 \sin \left (3 t \right ) c_{2} +3 \cos \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 69

DSolve[{D[x[t],t]==4*x[t]+5*y[t],D[y[t],t]==-2*x[t]+6*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{5 t} (3 c_1 \cos (3 t)-(c_1-5 c_2) \sin (3 t)) \\ y(t)\to \frac {1}{3} e^{5 t} (3 c_2 \cos (3 t)+(c_2-2 c_1) \sin (3 t)) \\ \end{align*}