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 }\left (t \right )&=x_{1} \left (t \right )+{\mathrm e}^{c t}\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 408
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)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{3} {\mathrm e}^{t}+\frac {{\mathrm e}^{c t}}{c -1} \\ x_{2} \left (t \right ) &= \frac {2 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c^{3}+2 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{1} c^{3}-3 c^{3} {\mathrm e}^{t} c_{3} \cos \left (2 t \right )-6 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c^{2}-6 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{1} c^{2}+9 c^{2} {\mathrm e}^{t} c_{3} \cos \left (2 t \right )-3 c^{3} {\mathrm e}^{t} c_{3} +14 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{2} c +14 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{1} c -21 \,{\mathrm e}^{t} c_{3} c \cos \left (2 t \right )+9 c^{2} {\mathrm e}^{t} c_{3} -10 c_{2} {\mathrm e}^{t} \sin \left (2 t \right )-10 c_{1} {\mathrm e}^{t} \cos \left (2 t \right )+15 \,{\mathrm e}^{t} c_{3} \cos \left (2 t \right )-21 \,{\mathrm e}^{t} c_{3} c +4 c \,{\mathrm e}^{t +t \left (c -1\right )}+15 c_{3} {\mathrm e}^{t}-16 \,{\mathrm e}^{t +t \left (c -1\right )}}{2 \left (c -1\right ) \left (c^{2}-2 c +5\right )} \\ x_{3} \left (t \right ) &= \frac {2 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{1} c^{3}-3 c^{3} {\mathrm e}^{t} c_{3} \sin \left (2 t \right )-2 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{2} c^{3}-6 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{1} c^{2}+9 c^{2} {\mathrm e}^{t} c_{3} \sin \left (2 t \right )+6 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{2} c^{2}+2 c^{3} {\mathrm e}^{t} c_{3} +14 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_{1} c -21 \,{\mathrm e}^{t} c_{3} c \sin \left (2 t \right )-14 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_{2} c -6 c^{2} {\mathrm e}^{t} c_{3} -10 c_{1} {\mathrm e}^{t} \sin \left (2 t \right )+15 \sin \left (2 t \right ) {\mathrm e}^{t} c_{3} +10 c_{2} {\mathrm e}^{t} \cos \left (2 t \right )+14 \,{\mathrm e}^{t} c_{3} c -10 c_{3} {\mathrm e}^{t}+6 \,{\mathrm e}^{c t} c +2 \,{\mathrm e}^{c t}}{2 \left (c -1\right ) \left (c^{2}-2 c +5\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.496 (sec). Leaf size: 256
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 {e^t \left (-3 c^3 c_1+9 c^2 c_1+\left (c^3-3 c^2+7 c-5\right ) (3 c_1+2 c_2) \cos (2 t)+2 \left (c^3-3 c^2+7 c-5\right ) (c_1-c_3) \sin (2 t)+4 c e^{(c-1) t}-16 e^{(c-1) t}-21 c c_1+15 c_1\right )}{2 (c-1) \left (c^2-2 c+5\right )} \\ \text {x3}(t)\to \frac {e^t \left (-2 \left (c^3-3 c^2+7 c-5\right ) (c_1-c_3) \cos (2 t)+\left (c^3-3 c^2+7 c-5\right ) (3 c_1+2 c_2) \sin (2 t)+2 \left (c^3-3 c^2+7 c-5\right ) c_1+2 (3 c+1) e^{(c-1) t}\right )}{2 (c-1) \left (c^2-2 c+5\right )} \\ \end{align*}