Internal problem ID [1640]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 10 Linear system of Differential equations. Section 10.6, constant coefficient
homogeneous system III. Page 566
Problem number: section 10.6, problem 5.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.282 (sec). Leaf size: 1975
dsolve([diff(y__1(t),t)=-3*y__1(t)-3*y__2(t)+1*y__3(t),diff(y__2(t),t)=0*y__1(t)+2*y__2(t)+2*y__3(t),diff(y__3(t),t)=5*y__1(t)+1*y__2(t)+1*y__3(t)],singsol=all)
\begin{align*} \text {Expression too large to display} \\ y_{2} \left (t \right ) &= {\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{3} +c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \\ y_{3} \left (t \right ) &= \frac {-2 c_{1} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}+c_{2} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{3} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-12 c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 187
DSolve[{y1'[t]==3*y1[t]-3*y2[t]+1*y3[t],y2'[t]==0*y1[t]+2*y2[t]+2*y3[t],y3'[t]==5*y1[t]+1*y2[t]+1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to \frac {1}{4} e^{-2 t} \left ((3 c_1-c_2+c_3) e^{6 t} \cos (2 t)+(c_1-3 c_2-c_3) e^{6 t} \sin (2 t)+c_1+c_2-c_3\right ) \\ \text {y2}(t)\to \frac {1}{4} e^{-2 t} \left (-(c_1-3 c_2-c_3) e^{6 t} \cos (2 t)+(3 c_1-c_2+c_3) e^{6 t} \sin (2 t)+c_1+c_2-c_3\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left ((c_1+c_2+c_3) e^{6 t} \cos (2 t)+2 (c_1-c_2) e^{6 t} \sin (2 t)-c_1-c_2+c_3\right ) \\ \end{align*}