6.22.15 problem section 10.5, problem 15

Internal problem ID [2268]
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 15
Date solved : Tuesday, September 30, 2025 at 05:25:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-3 y_{1} \left (t \right )-4 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-7 y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=2 \\ y_{2} \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 28

ode:=[diff(y__1(t),t) = -3*y__1(t)-4*y__2(t), diff(y__2(t),t) = y__1(t)-7*y__2(t)]; 
ic:=[y__1(0) = 2, y__2(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-5 t} \left (-8 t +2\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-5 t} \left (-16 t +12\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 30

ode={D[ y1[t],t]==-3*y1[t]-4*y2[t],D[ y2[t],t]==1*y1[t]-7*y2[t]}; 
ic={y1[0]==2,y2[0]==3}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^{-5 t} (2-8 t)\\ \text {y2}(t)&\to e^{-5 t} (3-4 t) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(3*y__1(t) + 4*y__2(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 7*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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