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 : Saturday, March 29, 2025 at 11:51:48 PM
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*}

✓ Maple. Time used: 0.118 (sec). Leaf size: 60

ode:=[diff(y__1(t),t) = 3*y__1(t)+5*y__2(t)+8*y__3(t), diff(y__2(t),t) = y__1(t)-y__2(t)-2*y__3(t), diff(y__3(t),t) = -y__1(t)-y__2(t)-y__3(t)]; 
dsolve(ode);
 

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

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

ode={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]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-3*y__1(t) - 5*y__2(t) - 8*y__3(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + y__2(t) + 2*y__3(t) + Derivative(y__2(t), t),0),Eq(y__1(t) + y__2(t) + y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - C_{1} e^{- 2 t} - \frac {2 C_{2} e^{t}}{3} - \frac {7 C_{3} e^{2 t}}{4}, \ y^{2}{\left (t \right )} = C_{1} e^{- 2 t} - \frac {4 C_{2} e^{t}}{3} - \frac {5 C_{3} e^{2 t}}{4}, \ y^{3}{\left (t \right )} = C_{2} e^{t} + C_{3} e^{2 t}\right ] \]