Internal problem ID [5962]
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: 5.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )-y \relax (t )+z \relax (t )+t -1\\ y^{\prime }\relax (t )&=2 x \relax (t )+y \relax (t )-z \relax (t )-3 t^{2}\\ z^{\prime }\relax (t )&=x \relax (t )+y \relax (t )+z \relax (t )+t^{2}-t +2 \end {align*}
✓ Solution by Maple
Time used: 0.126 (sec). Leaf size: 172
dsolve([diff(x(t),t)=x(t)-y(t)+z(t)+t-1,diff(y(t),t)=2*x(t)+y(t)-z(t)-3*t^2,diff(z(t),t)=x(t)+y(t)+z(t)+t^2-t+2],[x(t), y(t), z(t)], singsol=all)
\[ x \relax (t ) = t^{2}-\frac {1}{6}+\frac {2 c_{1} {\mathrm e}^{2 t}}{3}-c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )-c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \] \[ y \relax (t ) = -\frac {t^{2}}{2}-\frac {7}{4}+\frac {c_{1} {\mathrm e}^{2 t}}{3}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {3 t}{2} \] \[ z \relax (t ) = -\frac {3 t^{2}}{2}-\frac {t}{2}-\frac {7}{12}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )+c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \]
✓ Solution by Mathematica
Time used: 1.582 (sec). Leaf size: 1406
DSolve[{x'[t]==x[t]-y[t]+z[t]+t-1,y'[t]==2*x[t]+y[t]-z[t]-3*t^2,z'[t]==x[t]+y[t]+z[t]+t^2-t+2},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{55} e^{t/2} \left (\left (22 e^{3 t/2}-\sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )+33 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[1]} \left (22 \left (K[1]^2+1\right )-11 e^{\frac {3 K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[1]\right ) (K[1] (2 K[1]-5)+7)-\sqrt {11} e^{\frac {3 K[1]}{2}} (K[1] (34 K[1]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[1]\right )\right )dK[1]-10 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right ) \int _1^t\frac {1}{55} e^{-2 K[2]} \left (11 \left (K[2]^2+1\right )+e^{\frac {3 K[2]}{2}} \left (\sqrt {11} \left (28 K[2]^2-30 K[2]+43\right ) \sin \left (\frac {1}{2} \sqrt {11} K[2]\right )-11 \cos \left (\frac {1}{2} \sqrt {11} K[2]\right ) \left (16 K[2]^2+1\right )\right )\right )dK[2]+2 \left (11 e^{3 t/2}+2 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )-11 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[3]} \left (33 \left (K[3]^2+1\right )+11 e^{\frac {3 K[3]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[3]\right ) (K[3] (2 K[3]-5)+7)+\sqrt {11} e^{\frac {3 K[3]}{2}} (K[3] (34 K[3]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[3]\right )\right )dK[3]+22 (c_1+c_3) e^{3 t/2}+11 (3 c_1-2 c_3) \cos \left (\frac {\sqrt {11} t}{2}\right )-\sqrt {11} (c_1+10 c_2-4 c_3) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ y(t)\to \frac {1}{55} e^{t/2} \left (\left (11 e^{3 t/2}+17 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )-11 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[1]} \left (22 \left (K[1]^2+1\right )-11 e^{\frac {3 K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[1]\right ) (K[1] (2 K[1]-5)+7)-\sqrt {11} e^{\frac {3 K[1]}{2}} (K[1] (34 K[1]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[1]\right )\right )dK[1]+5 \left (\sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )+11 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[2]} \left (11 \left (K[2]^2+1\right )+e^{\frac {3 K[2]}{2}} \left (\sqrt {11} \left (28 K[2]^2-30 K[2]+43\right ) \sin \left (\frac {1}{2} \sqrt {11} K[2]\right )-11 \cos \left (\frac {1}{2} \sqrt {11} K[2]\right ) \left (16 K[2]^2+1\right )\right )\right )dK[2]+\left (11 e^{3 t/2}-13 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )-11 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[3]} \left (33 \left (K[3]^2+1\right )+11 e^{\frac {3 K[3]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[3]\right ) (K[3] (2 K[3]-5)+7)+\sqrt {11} e^{\frac {3 K[3]}{2}} (K[3] (34 K[3]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[3]\right )\right )dK[3]+11 (c_1+c_3) e^{3 t/2}-11 (c_1-5 c_2+c_3) \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (17 c_1+5 c_2-13 c_3) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ z(t)\to \frac {1}{55} e^{t/2} \left (\left (33 e^{3 t/2}+\sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )-33 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[1]} \left (22 \left (K[1]^2+1\right )-11 e^{\frac {3 K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[1]\right ) (K[1] (2 K[1]-5)+7)-\sqrt {11} e^{\frac {3 K[1]}{2}} (K[1] (34 K[1]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[1]\right )\right )dK[1]+10 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right ) \int _1^t\frac {1}{55} e^{-2 K[2]} \left (11 \left (K[2]^2+1\right )+e^{\frac {3 K[2]}{2}} \left (\sqrt {11} \left (28 K[2]^2-30 K[2]+43\right ) \sin \left (\frac {1}{2} \sqrt {11} K[2]\right )-11 \cos \left (\frac {1}{2} \sqrt {11} K[2]\right ) \left (16 K[2]^2+1\right )\right )\right )dK[2]+\left (33 e^{3 t/2}-4 \sqrt {11} \sin \left (\frac {\sqrt {11} t}{2}\right )+22 \cos \left (\frac {\sqrt {11} t}{2}\right )\right ) \int _1^t\frac {1}{55} e^{-2 K[3]} \left (33 \left (K[3]^2+1\right )+11 e^{\frac {3 K[3]}{2}} \cos \left (\frac {1}{2} \sqrt {11} K[3]\right ) (K[3] (2 K[3]-5)+7)+\sqrt {11} e^{\frac {3 K[3]}{2}} (K[3] (34 K[3]-5)+9) \sin \left (\frac {1}{2} \sqrt {11} K[3]\right )\right )dK[3]+33 (c_1+c_3) e^{3 t/2}+11 (2 c_3-3 c_1) \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (c_1+10 c_2-4 c_3) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ \end{align*}