90.35.2 problem 2

Internal problem ID [25503]
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 719
Problem number : 2
Date solved : Friday, October 03, 2025 at 12:02:13 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-3 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{1} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 33

ode:=[diff(y__1(t),t) = 5*y__1(t)-3*y__2(t), diff(y__2(t),t) = 2*y__1(t)]; 
ic:=[y__1(0) = 2, y__2(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

ode={D[y1[t],t]==5*y1[t]-3*y2[t], D[y2[t],t]==2*y1[t]}; 
ic={y1[0]==2,y2[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^{2 t} \left (3 e^t-1\right )\\ \text {y2}(t)&\to e^{2 t} \left (2 e^t-1\right ) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-5*y1(t) + 3*y2(t) + Derivative(y1(t), t),0),Eq(-2*y1(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 2, y2(0): 1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

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