Internal problem ID [5989]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.2. Page 346
Problem number: 15.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=\frac {9 x \relax (t )}{10}+\frac {21 y \relax (t )}{10}+\frac {16 z \relax (t )}{5}\\ y^{\prime }\relax (t )&=\frac {7 x \relax (t )}{10}+\frac {13 y \relax (t )}{2}+\frac {21 z \relax (t )}{5}\\ z^{\prime }\relax (t )&=\frac {11 x \relax (t )}{10}+\frac {17 y \relax (t )}{10}+\frac {17 z \relax (t )}{5} \end {align*}
✓ Solution by Maple
Time used: 0.209 (sec). Leaf size: 1014
dsolve([diff(x(t),t)=9/10*x(t)+21/10*y(t)+32/10*z(t),diff(y(t),t)=7/10*x(t)+65/10*y(t)+42/10*z(t),diff(z(t),t)=11/10*x(t)+17/10*y(t)+34/10*z(t)],[x(t), y(t), z(t)], singsol=all)
\[ x \relax (t ) = \frac {\left (-17 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+17 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+12420000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-124200 i \sqrt {29760999}-70975800 i \sqrt {3}-55740 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+372600 \sqrt {9920333}-12420000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-70975800\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{197820 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (-17 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-17 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+12420000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+124200 i \sqrt {29760999}-70975800 i \sqrt {3}+55740 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+372600 \sqrt {9920333}+12420000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+70975800\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{197820 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (-17 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+124200 i \sqrt {29760999}-27870 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+12420000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+70975800\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{98910 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \] \[ y \relax (t ) = \frac {\left (11 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-11 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+3600000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-36000 i \sqrt {29760999}-593964000 i \sqrt {3}+59340 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108000 \sqrt {9920333}-3600000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-593964000\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{197820 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (11 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+11 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+3600000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+36000 i \sqrt {29760999}-593964000 i \sqrt {3}-59340 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108000 \sqrt {9920333}+3600000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+593964000\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{197820 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (11 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+36000 i \sqrt {29760999}+29670 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+3600000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+593964000\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{98910 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \] \[ z \relax (t ) = c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}} \]
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 616
DSolve[{x'[t]==9/10*x[t]+21/10*y[t]+32/10*z[t],y'[t]==7/10*x[t]+65/10*y[t]+42/10*z[t],z'[t]==11/10*x[t]+17/10*y[t]+34/10*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 2 c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {16 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-599 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {21 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-170 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-99 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+1496 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ y(t)\to 7 c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+32 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+14 c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {3 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-11 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-43 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-46 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ z(t)\to c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {11 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-596 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {17 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+78 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-74 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+438 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ \end{align*}