Internal problem ID [1838]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number: 3.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1}\relax (t )\\ x_{2}^{\prime }\relax (t )&=3 x_{1}\relax (t )+x_{2}\relax (t )-2 x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=2 x_{1}\relax (t )+2 x_{2}\relax (t )+x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.068 (sec). Leaf size: 55
dsolve([diff(x__1(t),t)=1*x__1(t)-0*x__2(t)+0*x__3(t),diff(x__2(t),t)=3*x__1(t)+1*x__2(t)-2*x__3(t),diff(x__3(t),t)=2*x__1(t)+2*x__2(t)+1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = \frac {2 c_{1} {\mathrm e}^{t}}{3} \] \[ x_{2}\relax (t ) = \frac {{\mathrm e}^{t} \left (3 c_{2} \cos \left (2 t \right )-3 c_{3} \sin \left (2 t \right )-2 c_{1}\right )}{3} \] \[ x_{3}\relax (t ) = {\mathrm e}^{t} \left (c_{2} \sin \left (2 t \right )+c_{3} \cos \left (2 t \right )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 91
DSolve[{x1'[t]==1*x1[t]-0*x2[t]+0*x3[t],x2'[t]==3*x1[t]+1*x2[t]-2*x3[t],x3'[t]==2*x1[t]+2*x2[t]+1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to \frac {1}{2} e^t (2 (c_1+c_2) \cos (2 t)+(3 c_1-2 c_3) \sin (2 t)-2 c_1) \\ \text {x3}(t)\to \frac {1}{2} e^t ((2 c_3-3 c_1) \cos (2 t)+2 (c_1+c_2) \sin (2 t)+3 c_1) \\ \end{align*}