Internal problem ID [1855]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of
parameters. Page 366
Problem number: Example 2, page 364.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )+{\mathrm e}^{c t}\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )-2 x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{1} \relax (t )+2 x_{2} \relax (t )+x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 225
dsolve([diff(x__1(t),t)=1*x__1(t)+0*x__2(t)+0*x__3(t)+exp(c*t),diff(x__2(t),t)=2*x__1(t)+1*x__2(t)-2*x__3(t),diff(x__3(t),t)=3*x__1(t)+2*x__2(t)+1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {{\mathrm e}^{t} c_{1} c -c_{1} {\mathrm e}^{t}+{\mathrm e}^{c t}}{c -1} \] \[ x_{2} \relax (t ) = \frac {2 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{3} c^{3}-2 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c^{3}-6 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{3} c^{2}+6 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c^{2}-3 \,{\mathrm e}^{t} c_{1} c^{3}+14 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{3} c -14 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c +9 \,{\mathrm e}^{t} c_{1} c^{2}-10 c_{3} {\mathrm e}^{t} \cos \left (2 t \right )+10 c_{2} {\mathrm e}^{t} \sin \left (2 t \right )-21 \,{\mathrm e}^{t} c_{1} c +15 c_{1} {\mathrm e}^{t}+4 c \,{\mathrm e}^{c t}-16 \,{\mathrm e}^{c t}}{2 \left (c -1\right ) \left (c^{2}-2 c +5\right )} \] \[ x_{3} \relax (t ) = \frac {3 c \,{\mathrm e}^{c t}+{\mathrm e}^{c t}}{c^{3}-3 c^{2}+7 c -5}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{t} \cos \left (2 t \right )+c_{3} {\mathrm e}^{t} \sin \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.262 (sec). Leaf size: 162
DSolve[{x1'[t]==1*x1[t]+0*x2[t]+0*x3[t]+Exp[c*t],x2'[t]==2*x1[t]+1*x2[t]-2*x3[t],x3'[t]==3*x1[t]+2*x2[t]+1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^t \left (\frac {e^{(c-1) t}}{c-1}+c_1\right ) \\ \text {x2}(t)\to \frac {1}{2} e^t \left (\frac {4 (c-4) e^{(c-1) t}}{(c-1) ((c-2) c+5)}+(3 c_1+2 c_2) \cos (2 t)+2 (c_1-c_3) \sin (2 t)-3 c_1\right ) \\ \text {x3}(t)\to \frac {1}{2} e^t \left (\frac {(6 c+2) e^{(c-1) t}}{(c-1) ((c-2) c+5)}+2 (c_3-c_1) \cos (2 t)+(3 c_1+2 c_2) \sin (2 t)+2 c_1\right ) \\ \end{align*}