90.34.3 problem 3

Internal problem ID [25487]
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 677
Problem number : 3
Date solved : Friday, October 03, 2025 at 12:02:02 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.062 (sec). Leaf size: 24

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

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