90.33.8 problem 8

Internal problem ID [25482]
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 : 8
Date solved : Friday, October 03, 2025 at 12:02:00 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 20

ode:=[diff(y__1(t),t) = 3*y__1(t)-y__2(t), diff(y__2(t),t) = 4*y__1(t)-y__2(t)]; 
ic:=[y__1(0) = 1, y__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{t} \left (2 t +1\right ) \\ y_{2} \left (t \right ) &= 4 \,{\mathrm e}^{t} t \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 23

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

\begin{align*} \text {y1}(t)&\to e^t (2 t+1)\\ \text {y2}(t)&\to 4 e^t t \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-3*y1(t) + y2(t) + Derivative(y1(t), t),0),Eq(-4*y1(t) + y2(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 1, y2(0): 0} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = 2 t e^{t} + e^{t}, \ y_{2}{\left (t \right )} = 4 t e^{t}\right ] \]