Internal problem ID [353]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number: problem 50.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=9 x_{1}\relax (t )+13 x_{2}\relax (t )-13 x_{6}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-14 x_{1}\relax (t )+19 x_{2}\relax (t )-10 x_{3}\relax (t )-20 x_{4}\relax (t )+10 x_{5}\relax (t )+4 x_{6}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-30 x_{1}\relax (t )+12 x_{2}\relax (t )-7 x_{3}\relax (t )-30 x_{4}\relax (t )+12 x_{5}\relax (t )+18 x_{6}\relax (t )\\ x_{4}^{\prime }\relax (t )&=-12 x_{1}\relax (t )+10 x_{2}\relax (t )-10 x_{3}\relax (t )-9 x_{4}\relax (t )+10 x_{5}\relax (t )+2 x_{6}\relax (t )\\ x_{5}^{\prime }\relax (t )&=6 x_{1}\relax (t )+9 x_{2}\relax (t )+6 x_{4}\relax (t )+5 x_{5}\relax (t )-15 x_{6}\relax (t )\\ x_{6}^{\prime }\relax (t )&=-14 x_{1}\relax (t )+23 x_{2}\relax (t )-10 x_{3}\relax (t )-20 x_{4}\relax (t )+10 x_{5}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 135
dsolve([diff(x__1(t),t)=9*x__1(t)+13*x__2(t)+0*x__3(t)+0*x__4(t)+0*x__5(t)-13*x__6(t),diff(x__2(t),t)=-14*x__1(t)+19*x__2(t)-10*x__3(t)-20*x__4(t)+10*x__5(t)+4*x__6(t),diff(x__3(t),t)=-30*x__1(t)+12*x__2(t)-7*x__3(t)-30*x__4(t)+12*x__5(t)+18*x__6(t),diff(x__4(t),t)=-12*x__1(t)+10*x__2(t)-10*x__3(t)-9*x__4(t)+10*x__5(t)+2*x__6(t),diff(x__5(t),t)=6*x__1(t)+9*x__2(t)+0*x__3(t)+6*x__4(t)+5*x__5(t)-15*x__6(t),diff(x__6(t),t)=-14*x__1(t)+23*x__2(t)-10*x__3(t)-20*x__4(t)+10*x__5(t)+0*x__6(t)],[x__1(t), x__2(t), x__3(t), x__4(t), x__5(t), x__6(t)], singsol=all)
\[ x_{1}\relax (t ) = c_{3} {\mathrm e}^{-4 t}+c_{6} {\mathrm e}^{9 t} \] \[ x_{2}\relax (t ) = c_{4} {\mathrm e}^{-7 t}+c_{5} {\mathrm e}^{3 t}+c_{6} {\mathrm e}^{9 t} \] \[ x_{3}\relax (t ) = c_{4} {\mathrm e}^{-7 t}+c_{2} {\mathrm e}^{5 t}-{\mathrm e}^{11 t} c_{1} \] \[ x_{4}\relax (t ) = {\mathrm e}^{11 t} c_{1}+c_{4} {\mathrm e}^{-7 t}+c_{5} {\mathrm e}^{3 t} \] \[ x_{5}\relax (t ) = c_{2} {\mathrm e}^{5 t}+{\mathrm e}^{11 t} c_{1}+c_{3} {\mathrm e}^{-4 t} \] \[ x_{6}\relax (t ) = c_{3} {\mathrm e}^{-4 t}+c_{4} {\mathrm e}^{-7 t}+c_{5} {\mathrm e}^{3 t}+c_{6} {\mathrm e}^{9 t} \]
✓ Solution by Mathematica
Time used: 0.132 (sec). Leaf size: 1669
DSolve[{x1'[t]==9*x1[t]+13*x2[t]-13*x6[t],x2'[t]==-14*x1[t]+19*x2[t]-10*x3[t]-20*x4[t]+10*x5[t]+4*x6[t],x3'[t]==-30*x1[t]+12*x2[t]-7*x3[t]-30*x4[t]+12*x5[t]+18*x6[t],x4'[t]==-12*x1[t]+10*x2[t]-10*x3[t]-9*x4[t]+10*x5[t]+2*x6[t],x5'[t]==6*x1[t]+9*x2[t]+6*x4[t]+5*x5[t]-15*x6[t],x6'[t]==-14*x1[t]+23*x2[t]-10*x3[t]-20*x4[t]-10*x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t],x6[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {e^{\frac {1}{2} \left (7-5 \sqrt {57}\right ) t} \left (13 \left (6 c_1 \left (\left (665+243 \sqrt {57}\right ) e^{5 \sqrt {57} t}+15485 e^{\frac {1}{2} \left (3+5 \sqrt {57}\right ) t}+665-243 \sqrt {57}\right )-92910 (c_2-c_4+c_5) e^{\frac {1}{2} \left (3+5 \sqrt {57}\right ) t}+\left (9 \left (251 \sqrt {57}-817\right ) c_2+6 \left (665+243 \sqrt {57}\right ) c_4+40 \left (741-4 \sqrt {57}\right ) c_5+177 \left (19-21 \sqrt {57}\right ) c_6\right ) e^{5 \sqrt {57} t}-9 \left (817+251 \sqrt {57}\right ) c_2+6 \left (665-243 \sqrt {57}\right ) c_4+40 \left (741+4 \sqrt {57}\right ) c_5+177 \left (19+21 \sqrt {57}\right ) c_6\right )-6726 (32 c_1+13 (-16 c_2+15 c_4-5 c_5+c_6)) e^{\frac {1}{2} \left (11+5 \sqrt {57}\right ) t}\right )}{1096338} \\ \text {x2}(t)\to \frac {e^{-\left (\left (7+\frac {5 \sqrt {57}}{2}\right ) t\right )} \left (7119600390 (c_1-c_2+c_4-c_5) e^{\frac {5 \sqrt {57} t}{2}+12 t}-22474929 (477 c_1-449 c_2+89 c_3+388 c_4-369 c_5-28 c_6) e^{\frac {5}{2} \left (4+\sqrt {57}\right ) t}+48787041 (77 c_1-86 c_2+41 c_3+77 c_4-26 c_5+9 c_6) e^{\frac {5 \sqrt {57} t}{2}}-5 \left (3 \left (570599 \sqrt {57}-7900979\right ) c_1+18 \left (443049 \sqrt {57}-3732341\right ) c_2+3 \left (570599 \sqrt {57}-7900979\right ) c_4+10 \left (7028385-1201597 \sqrt {57}\right ) c_5+177 \left (513475-54727 \sqrt {57}\right ) c_6\right ) e^{\left (\frac {21}{2}+5 \sqrt {57}\right ) t}+5 \left (3 \left (7900979+570599 \sqrt {57}\right ) c_1+18 \left (3732341+443049 \sqrt {57}\right ) c_2+3 \left (7900979+570599 \sqrt {57}\right ) c_4-10 \left (7028385+1201597 \sqrt {57}\right ) c_5-177 \left (513475+54727 \sqrt {57}\right ) c_6\right ) e^{21 t/2}-12271587 (32 c_1+13 (-16 c_2+15 c_4-5 c_5+c_6)) e^{\frac {5 \sqrt {57} t}{2}+16 t}\right )}{2000268681} \\ \text {x3}(t)\to \frac {e^{-\left (\left (7+\frac {5 \sqrt {57}}{2}\right ) t\right )} \left (-162032 (c_1-c_2+c_4-c_5) e^{\frac {5 \sqrt {57} t}{2}+12 t}+2242 (77 c_1-86 c_2+41 c_3+77 c_4-26 c_5+9 c_6) e^{\frac {5 \sqrt {57} t}{2}}-3 \left (\left (1767+821 \sqrt {57}\right ) c_1+6 \left (227 \sqrt {57}-855\right ) c_2+\left (1767+821 \sqrt {57}\right ) c_4+10 \left (1729-29 \sqrt {57}\right ) c_5+59 \left (57-37 \sqrt {57}\right ) c_6\right ) e^{\left (\frac {21}{2}+5 \sqrt {57}\right ) t}+3 \left (\left (821 \sqrt {57}-1767\right ) c_1+6 \left (855+227 \sqrt {57}\right ) c_2+\left (821 \sqrt {57}-1767\right ) c_4-10 \left (1729+29 \sqrt {57}\right ) c_5-59 \left (57+37 \sqrt {57}\right ) c_6\right ) e^{21 t/2}\right )}{91922} \\ \text {x4}(t)\to \frac {e^{-\left (\left (7+\frac {5 \sqrt {57}}{2}\right ) t\right )} \left (81117270 (c_1-c_2+c_4-c_5) e^{\frac {5 \sqrt {57} t}{2}+12 t}-275766 (477 c_1-449 c_2+89 c_3+388 c_4-369 c_5-28 c_6) e^{\frac {5}{2} \left (4+\sqrt {57}\right ) t}+598614 (77 c_1-86 c_2+41 c_3+77 c_4-26 c_5+9 c_6) e^{\frac {5 \sqrt {57} t}{2}}+\left (3 \left (721639+2357 \sqrt {57}\right ) c_1+18 \left (243865-21741 \sqrt {57}\right ) c_2+3 \left (721639+2357 \sqrt {57}\right ) c_4+10 \left (89321 \sqrt {57}-253821\right ) c_5+177 \left (2171 \sqrt {57}-37031\right ) c_6\right ) e^{\left (\frac {21}{2}+5 \sqrt {57}\right ) t}-\left (3 \left (2357 \sqrt {57}-721639\right ) c_1-18 \left (243865+21741 \sqrt {57}\right ) c_2+3 \left (2357 \sqrt {57}-721639\right ) c_4+10 \left (253821+89321 \sqrt {57}\right ) c_5+177 \left (37031+2171 \sqrt {57}\right ) c_6\right ) e^{21 t/2}\right )}{24543174} \\ \text {x5}(t)\to \frac {e^{\frac {1}{2} \left (7-5 \sqrt {57}\right ) t} \left (c_1 \left (\left (171+49 \sqrt {57}\right ) e^{5 \sqrt {57} t}-342 e^{\frac {1}{2} \left (3+5 \sqrt {57}\right ) t}+171-49 \sqrt {57}\right )+342 (c_2-c_4+c_5) e^{\frac {1}{2} \left (3+5 \sqrt {57}\right ) t}+\left (3 \left (23 \sqrt {57}-57\right ) c_2+\left (171+49 \sqrt {57}\right ) c_4+10 \left (95+\sqrt {57}\right ) c_5-118 \sqrt {57} c_6\right ) e^{5 \sqrt {57} t}-3 \left (57+23 \sqrt {57}\right ) c_2+\left (171-49 \sqrt {57}\right ) c_4-10 \left (\sqrt {57}-95\right ) c_5+118 \sqrt {57} c_6\right )}{2242} \\ \text {x6}(t)\to -\frac {e^{-\left (\left (7+\frac {5 \sqrt {57}}{2}\right ) t\right )} \left (-5198438380 (c_1-c_2+c_4-c_5) e^{\frac {5 \sqrt {57} t}{2}+12 t}+14983286 (477 c_1-449 c_2+89 c_3+388 c_4-369 c_5-28 c_6) e^{\frac {5}{2} \left (4+\sqrt {57}\right ) t}-32524694 (77 c_1-86 c_2+41 c_3+77 c_4-26 c_5+9 c_6) e^{\frac {5 \sqrt {57} t}{2}}+\left (6 \left (24501545+1347523 \sqrt {57}\right ) c_1+9 \left (24088523-1209849 \sqrt {57}\right ) c_2+6 \left (24501545+1347523 \sqrt {57}\right ) c_4+40 \left (206511+1278686 \sqrt {57}\right ) c_5+177 \left (15839 \sqrt {57}-2055401\right ) c_6\right ) e^{\left (\frac {21}{2}+5 \sqrt {57}\right ) t}+\left (\left (147009270-8085138 \sqrt {57}\right ) c_1+9 \left (24088523+1209849 \sqrt {57}\right ) c_2+6 \left (24501545-1347523 \sqrt {57}\right ) c_4+40 \left (206511-1278686 \sqrt {57}\right ) c_5-177 \left (2055401+15839 \sqrt {57}\right ) c_6\right ) e^{21 t/2}+8181058 (32 c_1+13 (-16 c_2+15 c_4-5 c_5+c_6)) e^{\frac {5 \sqrt {57} t}{2}+16 t}\right )}{1333512454} \\ \end{align*}