12.21.11 problem section 10.4, problem 11

Internal problem ID [2249]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 11
Date solved : Monday, January 27, 2025 at 05:44:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=12 y_{1} \left (t \right )-4 y_{2} \left (t \right )+10 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-6 y_{1} \left (t \right )+y_{2} \left (t \right )-7 y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 72

dsolve([diff(y__1(t),t)=1*y__1(t)-1*y__2(t)+2*y__3(t),diff(y__2(t),t)=12*y__1(t)-4*y__2(t)+10*y__3(t),diff(y__3(t),t)=-6*y__1(t)+1*y__2(t)-7*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-5 t}+{\mathrm e}^{-2 t} c_2 +c_3 \,{\mathrm e}^{-3 t} \\ y_{2} \left (t \right ) &= 3 c_1 \,{\mathrm e}^{-5 t}+{\mathrm e}^{-2 t} c_2 +2 c_3 \,{\mathrm e}^{-3 t} \\ y_{3} \left (t \right ) &= -\frac {3 c_1 \,{\mathrm e}^{-5 t}}{2}-{\mathrm e}^{-2 t} c_2 -c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 193

DSolve[{D[ y1[t],t]==1*y1[t]-1*y2[t]+2*y3[t],D[ y2[t],t]==12*y1[t]-4*y2[t]+10*y3[t],D[ y1[t],t]==-6*y1[t]+1*y2[t]-7*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {e^{-7 t/6} \left (71 (77 c_1-109 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )+\sqrt {71} (143 c_2-2479 c_1) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{340800} \\ \text {y2}(t)\to \frac {e^{-7 t/6} \left (71 (2071 c_1-407 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )-\sqrt {71} (2717 c_1+5411 c_2) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{852000} \\ \text {y3}(t)\to \frac {e^{-7 t/6} \left (639 (23 c_1+9 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )+3 \sqrt {71} (937 c_1-329 c_2) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{568000} \\ \end{align*}