Internal problem ID [277]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 7.2, Matrices and Linear systems. Page 417
Problem number: problem 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right )+1\\ x_{2}^{\prime }\left (t \right )&=x_{3} \left (t \right )+x_{4} \left (t \right )+t\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{4} \left (t \right )+t^{2}\\ x_{4}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+t^{3} \end {align*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 273
dsolve([diff(x__1(t),t)=x__2(t)+x__3(t)+1,diff(x__2(t),t)=x__3(t)+x__4(t)+t,diff(x__3(t),t)=x__1(t)+x__4(t)+t^2,diff(x__4(t),t)=x__1(t)+x__2(t)+t^3],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \left (t \right ) = -\frac {11 t}{16}+\frac {t^{2}}{16}-\frac {t^{4}}{16}-c_{4} +\frac {c_{1} {\mathrm e}^{2 t}}{2}+\frac {c_{2} {\mathrm e}^{-t} \cos \left (t \right )}{2}+\frac {c_{2} \sin \left (t \right ) {\mathrm e}^{-t}}{2}-\frac {c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{2}+\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{2}-\frac {7 t^{3}}{24}+\frac {5}{16} \] \[ x_{2} \left (t \right ) = -\frac {3 t}{16}-\frac {3}{2}+\frac {t^{2}}{16}+\frac {t^{4}}{16}+c_{4} +\frac {c_{1} {\mathrm e}^{2 t}}{2}+\frac {c_{2} {\mathrm e}^{-t} \cos \left (t \right )}{2}-\frac {c_{2} \sin \left (t \right ) {\mathrm e}^{-t}}{2}+\frac {c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{2}+\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{2}-\frac {11 t^{3}}{24} \] \[ x_{3} \left (t \right ) = \frac {5 t}{16}-\frac {15 t^{2}}{16}-\frac {t^{4}}{16}-c_{4} +\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {c_{2} {\mathrm e}^{-t} \cos \left (t \right )}{2}-\frac {c_{2} \sin \left (t \right ) {\mathrm e}^{-t}}{2}+\frac {c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{2}-\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{2}+\frac {5 t^{3}}{24}-\frac {3}{16} \] \[ x_{4} \left (t \right ) = \frac {t^{3}}{24}-\frac {7 t^{2}}{16}+\frac {t^{4}}{16}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {c_{2} {\mathrm e}^{-t} \cos \left (t \right )}{2}+\frac {c_{2} \sin \left (t \right ) {\mathrm e}^{-t}}{2}-\frac {c_{3} {\mathrm e}^{-t} \cos \left (t \right )}{2}-\frac {c_{3} {\mathrm e}^{-t} \sin \left (t \right )}{2}-\frac {19 t}{16}+c_{4} \]
✓ Solution by Mathematica
Time used: 1.491 (sec). Leaf size: 442
DSolve[{x1'[t]==x2[t]+x3[t]+1,x2'[t]==x3[t]+x4[t]+t,x3'[t]==x1[t]+x4[t]+t^2,x4'[t]==x1[t]+x2[t]+t^3},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (-6 t^4-28 t^3+6 t^2-66 t+3 \left (8 c_1 \left (e^{2 t}+1\right )+8 c_2 \left (e^{2 t}-1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-3+8 c_3-8 c_4\right )\right )+48 (c_1-c_3) \cos (t)+48 (c_2-c_4) \sin (t)\right ) \text {x2}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (6 t^4-44 t^3+6 t^2-18 t+3 \left (8 c_1 \left (e^{2 t}-1\right )+8 c_2 \left (e^{2 t}+1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-35-8 c_3+8 c_4\right )\right )+48 (c_2-c_4) \cos (t)-48 (c_1-c_3) \sin (t)\right ) \text {x3}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (-6 t^4+20 t^3-90 t^2+30 t+3 \left (8 c_1 \left (e^{2 t}+1\right )+8 c_2 \left (e^{2 t}-1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}-19+8 c_3-8 c_4\right )\right )-48 (c_1-c_3) \cos (t)-48 (c_2-c_4) \sin (t)\right ) \text {x4}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (6 t^4+4 t^3-42 t^2-114 t+3 \left (8 c_1 \left (e^{2 t}-1\right )+8 c_2 \left (e^{2 t}+1\right )+8 c_3 e^{2 t}+8 c_4 e^{2 t}+13-8 c_3+8 c_4\right )\right )-48 (c_2-c_4) \cos (t)+48 (c_1-c_3) \sin (t)\right ) \end{align*}