Internal problem ID [5785]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page
400
Problem number: 3(f).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=-x \relax (t )+y \relax (t )-z \relax (t )\\ y^{\prime }\relax (t )&=2 x \relax (t )-y \relax (t )-4 z \relax (t )\\ z^{\prime }\relax (t )&=3 x \relax (t )-y \relax (t )+z \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 2595
dsolve([diff(x(t),t)=-x(t)+y(t)-z(t),diff(y(t),t)=2*x(t)-y(t)-4*z(t),diff(z(t),t)=3*x(t)-y(t)+z(t)],[x(t), y(t), z(t)], singsol=all)
\[ \text {Expression too large to display} \] \[ \text {Expression too large to display} \] \[ z \relax (t ) = c_{1} {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{-\frac {\left (-13-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (-13-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 501
DSolve[{x'[t]==-x[t]+y[t]-z[t],y'[t]==2*x[t]-y[t]-4*z[t],z'[t]==3*x[t]-y[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ y(t)\to 2 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-2 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {2 \text {$\#$1} e^{\text {$\#$1} t}+3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ z(t)\to -c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {3 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 \text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ \end{align*}