28.4.43 problem 7.43

Internal problem ID [4575]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.43
Date solved : Monday, January 27, 2025 at 09:23:41 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 47

dsolve([diff(x__1(t),t)=x__1(t)+x__2(t)+exp(2*t),diff(x__2(t),t)=-2*x__1(t)+3*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 54

DSolve[{D[x1[t],t]==x1[t]+x2[t]+Exp[2*t],D[x2[t],t]==-2x1[t]+3*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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