48.5.4 problem Problem 5.4

Internal problem ID [7569]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 5. Systems of First Order Differential Equations. Section 5.11 Problems. Page 360
Problem number : Problem 5.4
Date solved : Monday, January 27, 2025 at 03:06:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )+2 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 57

dsolve([diff(x__1(t),t)=4*x__1(t)-x__2(t),diff(x__2(t),t)=5*x__1(t)+2*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 70

DSolve[{D[ x1[t],t]==4*x1[t]-x2[t],D[ x2[t],t]==5*x1[t]+2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{3 t} (2 c_1 \cos (2 t)+(c_1-c_2) \sin (2 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{3 t} (2 c_2 \cos (2 t)+(5 c_1-c_2) \sin (2 t)) \\ \end{align*}