Internal problem ID [1829]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page
339
Problem number: 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )+6 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+9 x_{3} \left (t \right )+18 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )+10 x_{2} \left (t \right )+15 x_{3} \left (t \right )+30 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=7 x_{1} \left (t \right )+14 x_{2} \left (t \right )+21 x_{3} \left (t \right )+42 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 63
dsolve([diff(x__1(t),t)=1*x__1(t)+2*x__2(t)+3*x__3(t)+6*x__4(t),diff(x__2(t),t)=3*x__1(t)+6*x__2(t)+9*x__3(t)+18*x__4(t),diff(x__3(t),t)=5*x__1(t)+10*x__2(t)+15*x__3(t)+30*x__4(t),diff(x__4(t),t)=7*x__1(t)+14*x__2(t)+21*x__3(t)+42*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{3} +c_{4} {\mathrm e}^{64 t} \\ x_{2} \left (t \right ) &= 3 c_{3} +3 c_{4} {\mathrm e}^{64 t}+c_{2} \\ x_{3} \left (t \right ) &= 5 c_{3} +5 c_{4} {\mathrm e}^{64 t}+c_{1} \\ x_{4} \left (t \right ) &= 7 c_{4} {\mathrm e}^{64 t}-\frac {11 c_{3}}{3}-\frac {c_{2}}{3}-\frac {c_{1}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.089 (sec). Leaf size: 554
DSolve[{x1'[t]==1*x1[t]+2*x2[t]+3*x3[t]+6*x4[t],x2'[t]==3*x1[t]+6*x2[t]+9*x3[t]+19*x4[t],x3'[t]==5*x1[t]+10*x2[t]+15*x3[t]+30*x4[t],x4'[t]==7*x1[t]+14*x2[t]+21*x3[t]+42*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {e^{-\sqrt {1038} t} \left (2076 (7 c_1-c_4) e^{\sqrt {1038} t}-\left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4-1038 c_4\right ) e^{32 t}+\left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4+1038 c_4\right ) e^{2 \left (16+\sqrt {1038}\right ) t}\right )}{14532} \\ \text {x2}(t)\to \frac {\left (7 \left (519+13 \sqrt {1038}\right ) c_1+14 \left (519+13 \sqrt {1038}\right ) c_2+273 \sqrt {1038} c_3+10899 c_3-389 \sqrt {1038} c_4-8304 c_4\right ) e^{-\left (\left (\sqrt {1038}-32\right ) t\right )}+\left (\left (3633-91 \sqrt {1038}\right ) c_1+\left (7266-182 \sqrt {1038}\right ) c_2-273 \sqrt {1038} c_3+10899 c_3+389 \sqrt {1038} c_4-8304 c_4\right ) e^{\left (32+\sqrt {1038}\right ) t}-1038 (7 c_1+21 c_3-16 c_4)}{14532} \\ \text {x3}(t)\to \frac {e^{-\sqrt {1038} t} \left (2076 (7 c_3-5 c_4) e^{\sqrt {1038} t}-5 \left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4-1038 c_4\right ) e^{32 t}+5 \left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4+1038 c_4\right ) e^{2 \left (16+\sqrt {1038}\right ) t}\right )}{14532} \\ \text {x4}(t)\to \frac {e^{-\left (\left (\sqrt {1038}-32\right ) t\right )} \left (7 c_1 \left (e^{2 \sqrt {1038} t}-1\right )+14 c_2 \left (e^{2 \sqrt {1038} t}-1\right )+21 c_3 e^{2 \sqrt {1038} t}+\sqrt {1038} c_4 e^{2 \sqrt {1038} t}+10 c_4 e^{2 \sqrt {1038} t}-21 c_3+\sqrt {1038} c_4-10 c_4\right )}{2 \sqrt {1038}} \\ \end{align*}