6.22.24 problem section 10.5, problem 24

Internal problem ID [2277]
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 24
Date solved : Tuesday, September 30, 2025 at 05:25:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=5 y_{1} \left (t \right )-y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-y_{1} \left (t \right )+9 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )+4 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.171 (sec). Leaf size: 64

ode:=[diff(y__1(t),t) = 5*y__1(t)-y__2(t)+y__3(t), diff(y__2(t),t) = -y__1(t)+9*y__2(t)-3*y__3(t), diff(y__3(t),t) = -2*y__1(t)+2*y__2(t)+4*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{6 t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= \left (-2 c_3 \,t^{2}-4 c_2 t -3 c_3 t +c_1 \right ) {\mathrm e}^{6 t} \\ y_{3} \left (t \right ) &= {\mathrm e}^{6 t} \left (-2 c_3 \,t^{2}-4 c_2 t -2 c_3 t +c_1 +c_2 +c_3 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 106

ode={D[ y1[t],t]==5*y1[t]-1*y2[t]+1*y3[t],D[ y2[t],t]==-1*y1[t]+9*y2[t]-3*y3[t],D[ y3[t],t]==-2*y1[t]+2*y2[t]+4*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.114 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-5*y__1(t) + y__2(t) - y__3(t) + Derivative(y__1(t), t),0),Eq(y__1(t) - 9*y__2(t) + 3*y__3(t) + Derivative(y__2(t), t),0),Eq(2*y__1(t) - 2*y__2(t) - 4*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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