Internal problem ID [6715]
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 }\left (t \right )&=x \left (t \right )-y+z \left (t \right )+t -1\\ y^{\prime }&=2 x \left (t \right )+y-z \left (t \right )-3 t^{2}\\ z^{\prime }\left (t \right )&=x \left (t \right )+y+z \left (t \right )+t^{2}-t +2 \end {align*}
✓ Solution by Maple
Time used: 0.203 (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],singsol=all)
\begin{align*} x \left (t \right ) &= t^{2}-\frac {1}{6}+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 ) \\ y \left (t \right ) &= -\frac {t^{2}}{2}-\frac {7}{4}+\frac {c_{1} {\mathrm e}^{2 t}}{2}-\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 \left (t \right ) &= -\frac {t}{2}+\frac {3 c_{1} {\mathrm e}^{2 t}}{2}-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 )-\frac {3 t^{2}}{2}-\frac {7}{12} \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.906 (sec). Leaf size: 304
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 t^2+\frac {2}{5} c_1 e^{2 t}+\frac {2}{5} c_3 e^{2 t}+\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )-\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {1}{6} \\ y(t)\to \frac {1}{220} \left (-11 \left (10 t^2+30 t-4 (c_1+c_3) e^{2 t}+35\right )-44 (c_1-5 c_2+c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+4 \sqrt {11} (17 c_1+5 c_2-13 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ z(t)\to -\frac {3 t^2}{2}-\frac {t}{2}+\frac {3}{5} (c_1+c_3) e^{2 t}-\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {7}{12} \\ \end{align*}