10.16 problem 15

Internal problem ID [6742]

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 }\left (t \right )&=\frac {9 x \left (t \right )}{10}+\frac {21 y \left (t \right )}{10}+\frac {16 z \left (t \right )}{5}\\ y^{\prime }\left (t \right )&=\frac {7 x \left (t \right )}{10}+\frac {13 y \left (t \right )}{2}+\frac {21 z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {11 x \left (t \right )}{10}+\frac {17 y \left (t \right )}{10}+\frac {17 z \left (t \right )}{5} \end {align*}

✓ Solution by Maple

Time used: 0.109 (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)],singsol=all)
 

\begin{align*} x \left (t \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}}}}+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}}}} \\ y \left (t \right ) &= \frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-2070 i \sqrt {29760999}+6210 \sqrt {9920333}-207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930\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}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}-20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2070 i \sqrt {29760999}+207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6210 \sqrt {9920333}[phone]\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}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (16 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4140 i \sqrt {29760999}+414000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}[phone]\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}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ z \left (t \right ) &= -\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}-81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+35160 i \sqrt {29760999}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}-58655160\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}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}+81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}[phone]\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}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-40830 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}[phone]\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}}}}}{59154 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.029 (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*}