12.22.28 problem section 10.5, problem 28

Internal problem ID [2281]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.5, constant coefficient homogeneous system II. Page 555
Problem number : section 10.5, problem 28
Date solved : Tuesday, March 04, 2025 at 01:52:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-2 y_{1} \left (t \right )-12 y_{2} \left (t \right )+10 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=2 y_{1} \left (t \right )-24 y_{2} \left (t \right )+11 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=2 y_{1} \left (t \right )-24 y_{2} \left (t \right )+8 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 83

ode:=[diff(y__1(t),t) = -2*y__1(t)-12*y__2(t)+10*y__3(t), diff(y__2(t),t) = 2*y__1(t)-24*y__2(t)+11*y__3(t), diff(y__3(t),t) = 2*y__1(t)-24*y__2(t)+8*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-6 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-6 t} \left (18 c_3 \,t^{2}+18 c_2 t -24 c_3 t +18 c_1 -12 c_2 +5 c_3 \right )}{36} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-6 t} \left (6 c_3 \,t^{2}+6 c_2 t -6 c_3 t +6 c_1 -3 c_2 +c_3 \right )}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 131

ode={D[ y1[t],t]==-2*y1[t]-12*y2[t]+10*y3[t],D[ y2[t],t]==2*y1[t]-24*y2[t]+11*y3[t],D[ y3[t],t]==2*y1[t]-24*y2[t]+8*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-6 t} \left (c_1 \left (6 t^2+4 t+1\right )+2 t (c_3 (12 t+5)-6 c_2 (3 t+1))\right ) \\ \text {y2}(t)\to e^{-6 t} \left (-3 (c_1-6 c_2+4 c_3) t^2+(2 c_1-18 c_2+11 c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{-6 t} \left (-6 (c_1-6 c_2+4 c_3) t^2+2 (c_1-12 c_2+7 c_3) t+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(2*y__1(t) + 12*y__2(t) - 10*y__3(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) + 24*y__2(t) - 11*y__3(t) + Derivative(y__2(t), t),0),Eq(-2*y__1(t) + 24*y__2(t) - 8*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = 6 C_{1} t^{2} e^{- 6 t} + t \left (4 C_{1} + 12 C_{2}\right ) e^{- 6 t} + \left (C_{1} + 4 C_{2} + 12 C_{3}\right ) e^{- 6 t}, \ y^{2}{\left (t \right )} = - 3 C_{1} t^{2} e^{- 6 t} + t \left (2 C_{1} - 6 C_{2}\right ) e^{- 6 t} + \left (2 C_{2} - 6 C_{3}\right ) e^{- 6 t}, \ y^{3}{\left (t \right )} = - 6 C_{1} t^{2} e^{- 6 t} + t \left (2 C_{1} - 12 C_{2}\right ) e^{- 6 t} + \left (2 C_{2} - 12 C_{3}\right ) e^{- 6 t}\right ] \]