12.21.10 problem section 10.4, problem 10

Internal problem ID [2248]
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 10
Date solved : Monday, January 27, 2025 at 05:44:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=3 y_{1} \left (t \right )+5 y_{2} \left (t \right )+8 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-y_{1} \left (t \right )-y_{2} \left (t \right )-y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 60

dsolve([diff(y__1(t),t)=3*y__1(t)+5*y__2(t)+8*y__3(t),diff(y__2(t),t)=1*y__1(t)-1*y__2(t)-2*y__3(t),diff(y__3(t),t)=-1*y__1(t)-1*y__2(t)-1*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{-2 t} \\ y_{2} \left (t \right ) &= \frac {5 \,{\mathrm e}^{2 t} c_1}{7}+2 c_2 \,{\mathrm e}^{t}-c_3 \,{\mathrm e}^{-2 t} \\ y_{3} \left (t \right ) &= -\frac {4 \,{\mathrm e}^{2 t} c_1}{7}-\frac {3 c_2 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 193

DSolve[{D[ y1[t],t]==3*y1[t]+5*y2[t]+8*y3[t],D[ y2[t],t]==1*y1[t]-1*y2[t]-2*y3[t],D[ y1[t],t]==-1*y1[t]-1*y2[t]-1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {e^{-t/9} \left (\sqrt {35} (2 c_2-121 c_1) \sin \left (\frac {\sqrt {35} t}{9}\right )-7 (74 c_1+53 c_2) \cos \left (\frac {\sqrt {35} t}{9}\right )\right )}{1575} \\ \text {y2}(t)\to \frac {e^{-t/9} \left (7 (901 c_1+202 c_2) \cos \left (\frac {\sqrt {35} t}{9}\right )-\sqrt {35} (34 c_1+379 c_2) \sin \left (\frac {\sqrt {35} t}{9}\right )\right )}{4725} \\ \text {y3}(t)\to \frac {e^{-t/9} \left (2 \sqrt {35} (92 c_1+125 c_2) \sin \left (\frac {\sqrt {35} t}{9}\right )-14 (251 c_1+32 c_2) \cos \left (\frac {\sqrt {35} t}{9}\right )\right )}{4725} \\ \end{align*}