12.22.25 problem section 10.5, problem 25

Internal problem ID [2278]
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 25
Date solved : Saturday, March 29, 2025 at 11:52:32 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.123 (sec). Leaf size: 85

ode:=[diff(y__1(t),t) = y__1(t)+10*y__2(t)-12*y__3(t), diff(y__2(t),t) = 2*y__1(t)+2*y__2(t)+3*y__3(t), diff(y__3(t),t) = 2*y__1(t)-y__2(t)+6*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (6 c_3 \,t^{2}+6 c_2 t +6 c_3 t +6 c_1 +3 c_2 +4 c_3 \right )}{6} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (18 c_3 \,t^{2}+18 c_2 t +18 c_3 t +18 c_1 +9 c_2 +10 c_3 \right )}{18} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 118

ode={D[ y1[t],t]==1*y1[t]+10*y2[t]-12*y3[t],D[ y2[t],t]==2*y1[t]+2*y2[t]+3*y3[t],D[ y3[t],t]==2*y1[t]-1*y2[t]+6*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to -e^{3 t} (c_1 (2 t-1)+c_2 t (9 t-10)+3 c_3 (4-3 t) t) \\ \text {y2}(t)\to e^{3 t} \left (9 (c_2-c_3) t^2+(2 c_1-c_2+3 c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{3 t} \left (9 (c_2-c_3) t^2+(2 c_1-c_2+3 c_3) t+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.194 (sec). Leaf size: 114

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__1(t) - 10*y__2(t) + 12*y__3(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) - 2*y__2(t) - 3*y__3(t) + Derivative(y__2(t), t),0),Eq(-2*y__1(t) + y__2(t) - 6*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - 9 C_{3} t^{2} e^{3 t} - t \left (18 C_{2} - 10 C_{3}\right ) e^{3 t} - \left (18 C_{1} - 10 C_{2}\right ) e^{3 t}, \ y^{2}{\left (t \right )} = 9 C_{3} t^{2} e^{3 t} + t \left (18 C_{2} - C_{3}\right ) e^{3 t} + \left (18 C_{1} - C_{2} + C_{3}\right ) e^{3 t}, \ y^{3}{\left (t \right )} = 9 C_{3} t^{2} e^{3 t} + t \left (18 C_{2} - C_{3}\right ) e^{3 t} + \left (18 C_{1} - C_{2}\right ) e^{3 t}\right ] \]