Internal problem ID [5961]
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 8.1. Page 332
Problem number: 4.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )-y \relax (t )\\ y^{\prime }\relax (t )&=x \relax (t )+2 z \relax (t )\\ z^{\prime }\relax (t )&=-x \relax (t )+z \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.797 (sec). Leaf size: 2218
dsolve([diff(x(t),t)=x(t)-y(t),diff(y(t),t)=x(t)+2*z(t),diff(z(t),t)=-x(t)+z(t)],[x(t), y(t), z(t)], singsol=all)
\[ \text {Expression too large to display} \] \[ \text {Expression too large to display} \] \[ z \relax (t ) = -c_{2} {\mathrm e}^{\frac {\left (8-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {\left (8-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right )+c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 503
DSolve[{x'[t]==x[t]-y[t],y'[t]==x[t]+2*z[t],z'[t]==-x[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -2 c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]-c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\ y(t)\to c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-2 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\ z(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]-c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-\text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\ \end{align*}