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 }\relax (t )&=x_{2} \relax (t )+x_{3} \relax (t )+1\\ x_{2}^{\prime }\relax (t )&=x_{3} \relax (t )+x_{4} \relax (t )+t\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{4} \relax (t )+t^{2}\\ x_{4}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t )+t^{3} \end {align*}
✓ Solution by Maple
Time used: 0.125 (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)
\[ x_{1} \relax (t ) = -\frac {11 t}{16}-c_{4}+\frac {5}{16}-\frac {7 t^{3}}{24}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {t^{4}}{16}+\frac {t^{2}}{16}+\frac {c_{2} {\mathrm e}^{-t} \cos \relax (t )}{2}+\frac {c_{2} \sin \relax (t ) {\mathrm e}^{-t}}{2}-\frac {c_{3} {\mathrm e}^{-t} \cos \relax (t )}{2}+\frac {c_{3} {\mathrm e}^{-t} \sin \relax (t )}{2} \] \[ x_{2} \relax (t ) = -\frac {3}{2}-\frac {3 t}{16}+c_{4}-\frac {11 t^{3}}{24}+\frac {c_{1} {\mathrm e}^{2 t}}{2}+\frac {t^{4}}{16}+\frac {t^{2}}{16}+\frac {c_{2} {\mathrm e}^{-t} \cos \relax (t )}{2}-\frac {c_{2} \sin \relax (t ) {\mathrm e}^{-t}}{2}+\frac {c_{3} {\mathrm e}^{-t} \cos \relax (t )}{2}+\frac {c_{3} {\mathrm e}^{-t} \sin \relax (t )}{2} \] \[ x_{3} \relax (t ) = \frac {5 t}{16}-c_{4}+\frac {5 t^{3}}{24}-\frac {3}{16}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {t^{4}}{16}-\frac {15 t^{2}}{16}-\frac {c_{2} {\mathrm e}^{-t} \cos \relax (t )}{2}-\frac {c_{2} \sin \relax (t ) {\mathrm e}^{-t}}{2}+\frac {c_{3} {\mathrm e}^{-t} \cos \relax (t )}{2}-\frac {c_{3} {\mathrm e}^{-t} \sin \relax (t )}{2} \] \[ x_{4} \relax (t ) = \frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {c_{2} {\mathrm e}^{-t} \cos \relax (t )}{2}+\frac {c_{2} \sin \relax (t ) {\mathrm e}^{-t}}{2}-\frac {c_{3} {\mathrm e}^{-t} \cos \relax (t )}{2}-\frac {c_{3} {\mathrm e}^{-t} \sin \relax (t )}{2}-\frac {7 t^{2}}{16}+\frac {t^{3}}{24}+\frac {t^{4}}{16}-\frac {19 t}{16}+c_{4} \]
✓ Solution by Mathematica
Time used: 1.029 (sec). Leaf size: 369
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 (-2 t (t (t (3 t+14)-3)+33)+3 \left (8 (c_1+c_2+c_3+c_4) e^{2 t}-3+8 c_1-8 c_2+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 (2 t (t (t (3 t-22)+3)-9)+3 \left (8 (c_1+c_2+c_3+c_4) e^{2 t}-35-8 c_1+8 c_2-8 c_3+8 c_4\right )\right )+48 (c_2-c_4) \cos (t)+48 (c_3-c_1) \sin (t)\right ) \\ \text {x3}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (2 t (t ((10-3 t) t-45)+15)+3 \left (8 (c_1+c_2+c_3+c_4) e^{2 t}-19+8 c_1-8 c_2+8 c_3-8 c_4\right )\right )+48 (c_3-c_1) \cos (t)+48 (c_4-c_2) \sin (t)\right ) \\ \text {x4}(t)\to \frac {1}{96} e^{-t} \left (e^t \left (2 t (t (t+3) (3 t-7)-57)+3 \left (8 (c_1+c_2+c_3+c_4) e^{2 t}+13-8 c_1+8 c_2-8 c_3+8 c_4\right )\right )+48 (c_4-c_2) \cos (t)+48 (c_1-c_3) \sin (t)\right ) \\ \end{align*}