Internal problem ID [1839]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1}\relax (t )+x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{2}\relax (t )-x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-2 x_{1}\relax (t )-x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.088 (sec). Leaf size: 66
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)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = -\frac {c_{2} \cos \relax (t )}{2}+\frac {c_{3} \sin \relax (t )}{2}-\frac {\sin \relax (t ) c_{2}}{2}-\frac {c_{3} \cos \relax (t )}{2} \] \[ x_{2}\relax (t ) = \frac {\sin \relax (t ) c_{2}}{2}-\frac {c_{3} \sin \relax (t )}{2}+\frac {c_{2} \cos \relax (t )}{2}+\frac {c_{3} \cos \relax (t )}{2}+c_{1} {\mathrm e}^{t} \] \[ x_{3}\relax (t ) = \sin \relax (t ) c_{2}+c_{3} \cos \relax (t ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 67
DSolve[{x1'[t]==1*x1[t]-0*x2[t]+1*x3[t],x2'[t]==0*x1[t]+1*x2[t]-1*x3[t],x3'[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*}