90.33.7 problem 7

Internal problem ID [25481]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 645
Problem number : 7
Date solved : Friday, October 03, 2025 at 12:01:59 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-2 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=4 y_{1} \left (t \right )-y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 27

ode:=[diff(y__1(t),t) = 5*y__1(t)-2*y__2(t), diff(y__2(t),t) = 4*y__1(t)-y__2(t)]; 
ic:=[y__1(0) = 0, y__2(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 31

ode={D[y1[t],t]==5*y1[t]-2*y2[t], D[y2[t],t]==4*y1[t]-y2[t]}; 
ic={y1[0]==0,y2[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.062 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-5*y1(t) + 2*y2(t) + Derivative(y1(t), t),0),Eq(-4*y1(t) + y2(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 0, y2(0): 1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - e^{3 t} + e^{t}, \ y_{2}{\left (t \right )} = - e^{3 t} + 2 e^{t}\right ] \]