28.4.50 problem 7.50

Internal problem ID [4582]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.50
Date solved : Monday, January 27, 2025 at 09:23:48 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right )-2 x_{3} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )-6 x_{2} \left (t \right )+5 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 4.213 (sec). Leaf size: 753

dsolve([diff(x__1(t),t)=-3*x__1(t)+4*x__2(t)-2*x__3(t)+exp(t),diff(x__2(t),t)=x__1(t)+x__2(t),diff(x__3(t),t)=6*x__1(t)-6*x__2(t)+5*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {4 c_{1} \sqrt {6}\, {\mathrm e}^{\frac {\sqrt {6}\, \left (-8 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )+\sqrt {6}\right ) t}{6}} \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )}{3}-\frac {i {\mathrm e}^{-\frac {\left (i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}+9 \left (-54+6 i \sqrt {303}\right )^{{2}/{3}}-96 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}-288\right ) t}{288}} \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}\, c_{2}}{288}-\frac {{\mathrm e}^{-\frac {\left (i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}+9 \left (-54+6 i \sqrt {303}\right )^{{2}/{3}}-96 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}-288\right ) t}{288}} \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} c_{2}}{32}+\frac {{\mathrm e}^{-\frac {\left (i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}+9 \left (-54+6 i \sqrt {303}\right )^{{2}/{3}}-96 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}-288\right ) t}{288}} \left (-54+6 i \sqrt {303}\right )^{{1}/{3}} c_{2}}{3}-\frac {2 c_3 \sqrt {6}\, {\mathrm e}^{\frac {\sqrt {2}\, \left (-4 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}+12 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )\right ) t}{6}} \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )}{3}+2 \sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, \left (-4 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}+12 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )\right ) t}{6}} \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right ) c_3 \\ x_{2} \left (t \right ) &= -{\mathrm e}^{t}+c_{1} {\mathrm e}^{\frac {\sqrt {6}\, \left (-8 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )+\sqrt {6}\right ) t}{6}}+c_{2} {\mathrm e}^{-\frac {\left (i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}+9 \left (-54+6 i \sqrt {303}\right )^{{2}/{3}}-96 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}-288\right ) t}{288}}+c_3 \,{\mathrm e}^{\frac {\sqrt {2}\, \left (-4 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}+12 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )\right ) t}{6}} \\ x_{3} \left (t \right ) &= \left (\frac {4 c_3 \sqrt {6}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )}{3}-4 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right ) c_3 +\left (-\frac {2}{3}-\frac {4 \cos \left (\frac {2 \arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{3}\right )}{3}+\frac {4 \sqrt {3}\, \sin \left (\frac {2 \arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{3}\right )}{3}\right ) c_3 \right ) {\mathrm e}^{\frac {\sqrt {2}\, \left (-4 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}+12 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}+\frac {\pi }{6}\right )\right ) t}{6}}+\left (-\frac {2}{3}+\frac {37 \left (-54+6 i \sqrt {303}\right )^{{4}/{3}}}{27648}+\frac {\left (-54+6 i \sqrt {303}\right )^{{2}/{3}}}{144}-\frac {2 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}}{3}+\frac {i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}}{144}-\frac {i \left (-54+6 i \sqrt {303}\right )^{{4}/{3}} \sqrt {303}}{9216}\right ) c_{2} {\mathrm e}^{-\frac {\left (i \left (-54+6 i \sqrt {303}\right )^{{2}/{3}} \sqrt {303}+9 \left (-54+6 i \sqrt {303}\right )^{{2}/{3}}-96 \left (-54+6 i \sqrt {303}\right )^{{1}/{3}}-288\right ) t}{288}}+\left (\frac {8 c_{1} \sqrt {6}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )}{3}+\left (-\frac {2}{3}-\frac {8 \cos \left (\frac {2 \arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )}{3}\right ) c_{1} \right ) {\mathrm e}^{\frac {\sqrt {6}\, \left (-8 \cos \left (\frac {\arctan \left (\frac {\sqrt {303}}{9}\right )}{3}\right )+\sqrt {6}\right ) t}{6}}-\frac {3 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 1701

DSolve[{D[x1[t],t]==-3*x1[t]+4*x2[t]-2*x3[t]+Exp[t],D[x2[t],t]==x1[t]+x2[t],D[x3[t],t]==6*x1[t]-6*x2[t]+5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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