14.23.4 problem 4

Internal problem ID [2743]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 4
Date solved : Monday, January 27, 2025 at 06:12:36 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.093 (sec). Leaf size: 65

dsolve([diff(x__1(t),t)=1*x__1(t)-0*x__2(t)+1*x__3(t),diff(x__2(t),t)=0*x__1(t)+1*x__2(t)-1*x__3(t),diff(x__3(t),t)=-2*x__1(t)-0*x__2(t)-1*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {c_3 \sin \left (t \right )}{2}-\frac {c_2 \cos \left (t \right )}{2}-\frac {\sin \left (t \right ) c_2}{2}-\frac {\cos \left (t \right ) c_3}{2} \\ x_{2} \left (t \right ) &= \frac {\sin \left (t \right ) c_2}{2}-\frac {c_3 \sin \left (t \right )}{2}+\frac {c_2 \cos \left (t \right )}{2}+\frac {\cos \left (t \right ) c_3}{2}+{\mathrm e}^{t} c_1 \\ x_{3} \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 67

DSolve[{D[ x1[t],t]==1*x1[t]-0*x2[t]+1*x3[t],D[ x2[t],t]==0*x1[t]+1*x2[t]-1*x3[t],D[ x3[t],t]==-2*x1[t]-0*x2[t]-1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (t)+(c_1+c_3) \sin (t) \\ \text {x2}(t)\to (c_1+c_2) e^t-c_1 \cos (t)-(c_1+c_3) \sin (t) \\ \text {x3}(t)\to c_3 \cos (t)-(2 c_1+c_3) \sin (t) \\ \end{align*}