76.25.19 problem 23

Internal problem ID [17832]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 23
Date solved : Tuesday, January 28, 2025 at 11:03:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{2} \left (t \right )-2 x_{3} \left (t \right )+3 x_{4} \left (t \right )+2 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )+6 x_{2} \left (t \right )+4 x_{3} \left (t \right )-8 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-8 x_{1} \left (t \right )-8 x_{2} \left (t \right )-6 x_{3} \left (t \right )+8 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=8 x_{1} \left (t \right )+7 x_{2} \left (t \right )+4 x_{3} \left (t \right )-9 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=-3 x_{1} \left (t \right )-5 x_{2} \left (t \right )-3 x_{3} \left (t \right )+5 x_{4} \left (t \right )+7 x_{5} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.361 (sec). Leaf size: 396

dsolve([diff(x__1(t),t)=0*x__1(t)-3*x__2(t)-2*x__3(t)+3*x__4(t)+2*x__5(t),diff(x__2(t),t)=8*x__1(t)+6*x__2(t)+4*x__3(t)-8*x__4(t)-16*x__5(t),diff(x__3(t),t)=-8*x__1(t)-8*x__2(t)-6*x__3(t)+8*x__4(t)-16*x__5(t),diff(x__4(t),t)=8*x__1(t)+7*x__2(t)+4*x__3(t)-9*x__4(t)-16*x__5(t),diff(x__5(t),t)=-3*x__1(t)-5*x__2(t)-3*x__3(t)+5*x__4(t)+7*x__5(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}+c_4 \,{\mathrm e}^{\frac {\left (-1+\sqrt {393}\right ) t}{2}}+c_5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {393}\right ) t}{2}} \\ x_{2} \left (t \right ) &= -\frac {192 \left (-\sqrt {393}\, c_5 \,{\mathrm e}^{-2 t -\frac {t \left (-3+\sqrt {393}\right )}{2}}+\sqrt {393}\, c_4 \,{\mathrm e}^{-2 t +\frac {t \left (3+\sqrt {393}\right )}{2}}+256 c_{2} {\mathrm e}^{-t}-26 c_{1} {\mathrm e}^{-2 t}-26 c_{3} {\mathrm e}^{2 t}+61 c_5 \,{\mathrm e}^{-2 t -\frac {t \left (-3+\sqrt {393}\right )}{2}}+61 c_4 \,{\mathrm e}^{-2 t +\frac {t \left (3+\sqrt {393}\right )}{2}}\right )}{13 \left (3+\sqrt {393}\right ) \left (-3+\sqrt {393}\right )} \\ x_{3} \left (t \right ) &= -\frac {3 \sqrt {393}\, {\mathrm e}^{\frac {\left (-1+\sqrt {393}\right ) t}{2}} c_4}{13}+\frac {3 \sqrt {393}\, {\mathrm e}^{-\frac {\left (1+\sqrt {393}\right ) t}{2}} c_5}{13}+\frac {224 c_{2} {\mathrm e}^{-t}}{13}-c_{3} {\mathrm e}^{2 t}+\frac {25 c_4 \,{\mathrm e}^{\frac {\left (-1+\sqrt {393}\right ) t}{2}}}{13}+\frac {25 c_5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {393}\right ) t}{2}}}{13} \\ x_{4} \left (t \right ) &= \frac {\frac {192 \sqrt {393}\, c_5 \,{\mathrm e}^{-2 t -\frac {t \left (-3+\sqrt {393}\right )}{2}}}{13}-\frac {192 \sqrt {393}\, c_4 \,{\mathrm e}^{-2 t +\frac {t \left (3+\sqrt {393}\right )}{2}}}{13}+\frac {7296 c_{2} {\mathrm e}^{-t}}{13}+384 c_{1} {\mathrm e}^{-2 t}+384 c_{3} {\mathrm e}^{2 t}-\frac {11712 c_5 \,{\mathrm e}^{-2 t -\frac {t \left (-3+\sqrt {393}\right )}{2}}}{13}-\frac {11712 c_4 \,{\mathrm e}^{-2 t +\frac {t \left (3+\sqrt {393}\right )}{2}}}{13}}{\left (3+\sqrt {393}\right ) \left (-3+\sqrt {393}\right )} \\ x_{5} \left (t \right ) &= \frac {\sqrt {393}\, {\mathrm e}^{\frac {\left (-1+\sqrt {393}\right ) t}{2}} c_4}{52}-\frac {\sqrt {393}\, {\mathrm e}^{-\frac {\left (1+\sqrt {393}\right ) t}{2}} c_5}{52}-\frac {3 c_{2} {\mathrm e}^{-t}}{13}+\frac {87 c_4 \,{\mathrm e}^{\frac {\left (-1+\sqrt {393}\right ) t}{2}}}{52}+\frac {87 c_5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {393}\right ) t}{2}}}{52} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 408

DSolve[{D[x1[t],t]==0*x1[t]-3*x2[t]-2*x3[t]+3*x4[t]+2*x5[t],D[x2[t],t]==8*x1[t]+6*x2[t]+4*x3[t]-8*x4[t]-16*x5[t],D[x3[t],t]==-8*x1[t]-8*x2[t]-6*x3[t]+8*x4[t]+16*x5[t],D[x4[t],t]==8*x1[t]+7*x2[t]+4*x3[t]-9*x4[t]-16*x5[t],D[x5[t],t]==-3*x1[t]-5*x2[t]-3*x3[t]+5*x4[t]+7*x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} \left (c_1 \left (-2 e^{3 t}+2 e^{4 t}+1\right )+c_2 \left (e^t-4 e^{3 t}+2 e^{4 t}+1\right )-2 c_3 e^{3 t}+c_3 e^{4 t}-c_4 e^t+4 c_4 e^{3 t}-2 c_4 e^{4 t}+6 c_5 e^{3 t}-4 c_5 e^{4 t}+c_3-c_4-2 c_5\right ) \\ \text {x2}(t)\to e^{-2 t} \left (2 c_1 \left (e^{4 t}-1\right )+c_2 \left (2 e^{4 t}-1\right )+(c_3-2 c_4-4 c_5) \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to e^{-2 t} \left (-2 c_1 \left (e^{4 t}-1\right )-2 c_2 \left (e^{4 t}-1\right )-c_3 e^{4 t}+2 c_4 e^{4 t}+4 c_5 e^{4 t}+2 c_3-2 c_4-4 c_5\right ) \\ \text {x4}(t)\to e^{-2 t} \left (2 c_1 \left (e^{4 t}-1\right )+c_2 \left (-e^t+2 e^{4 t}-1\right )+c_3 e^{4 t}+c_4 e^t-2 c_4 e^{4 t}-4 c_5 e^{4 t}-c_3+2 c_4+4 c_5\right ) \\ \text {x5}(t)\to e^{-2 t} \left (-\left (c_1 \left (e^{3 t}-1\right )\right )+c_2 \left (e^t-2 e^{3 t}+1\right )-c_3 e^{3 t}-c_4 e^t+2 c_4 e^{3 t}+3 c_5 e^{3 t}+c_3-c_4-2 c_5\right ) \\ \end{align*}