12.22.14 problem section 10.5, problem 14

Internal problem ID [2267]
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 14
Date solved : Tuesday, March 04, 2025 at 01:52:43 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=15 y_{1} \left (t \right )-9 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=16 y_{1} \left (t \right )-9 y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 5\\ y_{2} \left (0\right ) = 8 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 28

ode:=[diff(y__1(t),t) = 15*y__1(t)-9*y__2(t), diff(y__2(t),t) = 16*y__1(t)-9*y__2(t)]; 
ic:=y__1(0) = 5y__2(0) = 8; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (-12 t +5\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (-144 t +72\right )}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 31

ode={D[ y1[t],t]==15*y1[t]-9*y2[t],D[ y2[t],t]==16*y1[t]-9*y2[t]}; 
ic={y1[0]==5,y2[0]==8}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{3 t} (5-12 t) \\ \text {y2}(t)\to -8 e^{3 t} (2 t-1) \\ \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-15*y__1(t) + 9*y__2(t) + Derivative(y__1(t), t),0),Eq(-16*y__1(t) + 9*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = 12 C_{1} t e^{3 t} + \left (C_{1} + 12 C_{2}\right ) e^{3 t}, \ y^{2}{\left (t \right )} = 16 C_{1} t e^{3 t} + 16 C_{2} e^{3 t}\right ] \]