52.10.16 problem 15

Internal problem ID [8410]
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
Date solved : Monday, January 27, 2025 at 03:57:41 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.133 (sec). Leaf size: 918

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 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}} \\ y &= -\frac {\left (-8 i \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}} \sqrt {3}+8 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}-207000 i \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}} \sqrt {3}+2070 i \sqrt {29760999}[phone] i \sqrt {3}-20757 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+207000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-6210 \sqrt {9920333}[phone]\right ) c_{1} {\mathrm e}^{-\frac {\left (i \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}}-\frac {\left (8 i \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}} \sqrt {3}+8 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}+207000 i \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}} \sqrt {3}+2070 i \sqrt {29760999}-299382930 i \sqrt {3}-20757 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+207000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6210 \sqrt {9920333}[phone]\right ) c_{2} {\mathrm e}^{\frac {\left (i \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}}+\frac {\left (16 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}+4140 i \sqrt {29760999}+20757 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+414000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}[phone]\right ) c_3 \,{\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}} \\ z \left (t \right ) &= -\frac {\left (7 i \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}} \sqrt {3}-7 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}-3516000 i \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}} \sqrt {3}+35160 i \sqrt {29760999}-58655160 i \sqrt {3}-81660 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-105480 \sqrt {9920333}-58655160\right ) c_{1} {\mathrm e}^{-\frac {\left (i \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}}-\frac {\left (-7 i \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}} \sqrt {3}-7 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}+3516000 i \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}} \sqrt {3}+35160 i \sqrt {29760999}[phone] i \sqrt {3}-81660 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+105480 \sqrt {9920333}-58655160\right ) c_{2} {\mathrm e}^{\frac {\left (i \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}} \sqrt {3}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}}+\frac {\left (-7 \left (329940+60 i \sqrt {29760999}\right )^{{4}/{3}}+35160 i \sqrt {29760999}+40830 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}-58655160\right ) c_3 \,{\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{{1}/{3}}}}}{59154 \left (329940+60 i \sqrt {29760999}\right )^{{2}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 616

DSolve[{D[x[t],t]==9/10*x[t]+21/10*y[t]+32/10*z[t],D[y[t],t]==7/10*x[t]+65/10*y[t]+42/10*z[t],D[z[t],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*}