Internal problem ID [1843]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number: 8.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{2}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-2 x_{1}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-3 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=3 x_{3}\relax (t ) \end {align*}
With initial conditions \[ [x_{1}\relax (0) = 1, x_{2}\relax (0) = 1, x_{3}\relax (0) = 1, x_{4}\relax (0) = 0] \]
✓ Solution by Maple
Time used: 0.064 (sec). Leaf size: 42
dsolve([diff(x__1(t),t) = 2*x__2(t), diff(x__2(t),t) = -2*x__1(t), diff(x__3(t),t) = -3*x__4(t), diff(x__4(t),t) = 3*x__3(t), x__1(0) = 1, x__2(0) = 1, x__3(0) = 1, x__4(0) = 0],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = \cos \left (2 t \right )+\sin \left (2 t \right ) \] \[ x_{2}\relax (t ) = -\sin \left (2 t \right )+\cos \left (2 t \right ) \] \[ x_{3}\relax (t ) = \cos \left (3 t \right ) \] \[ x_{4}\relax (t ) = \sin \left (3 t \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 42
DSolve[{x1'[t]==-0*x1[t]+2*x2[t]+0*x3[t]+0*x4[t],x2'[t]==-2*x1[t]-0*x2[t]-0*x3[t]+0*x4[t],x3'[t]==0*x1[t]-0*x2[t]-0*x3[t]-3*x4[t],x4'[t]==0*x1[t]-0*x2[t]+3*x3[t]-0*x4[t]},{x1[0]==1,x2[0]==1,x3[0]==1,x4[0]==0},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \sin (2 t)+\cos (2 t) \\ \text {x2}(t)\to \cos (2 t)-\sin (2 t) \\ \text {x3}(t)\to \cos (3 t) \\ \text {x4}(t)\to \sin (3 t) \\ \end{align*}