90.34.16 problem 16

Internal problem ID [25500]
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 : 16
Date solved : Friday, October 03, 2025 at 12:02:11 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 48

ode:=[diff(y__1(t),t) = -2*y__1(t)+2*y__2(t)+y__3(t)+exp(-2*t), diff(y__2(t),t) = -y__2(t), diff(y__3(t),t) = 2*y__1(t)-2*y__2(t)-y__3(t)-exp(-2*t)]; 
ic:=[y__1(0) = 2, y__2(0) = 1, y__3(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 58

ode={D[y1[t],t]==-2*y1[t]+2*y2[t]+y3[t]+Exp[-2*t], D[y2[t],t]==-y2[t],D[y3[t],t]==2*y1[t]-2*y2[t]-y3[t]-Exp[-2*t]}; 
ic={y1[0]==2,y2[0]==1,y3[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^{-3 t} \left (e^t+e^{2 t}+e^{3 t}-1\right )\\ \text {y2}(t)&\to e^{-t}\\ \text {y3}(t)&\to e^{-3 t}-e^{-2 t}-e^{-t}+2 \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
y3 = Function("y3") 
ode=[Eq(2*y1(t) - 2*y2(t) - y3(t) + Derivative(y1(t), t) - exp(-2*t),0),Eq(y2(t) + Derivative(y2(t), t),0),Eq(-2*y1(t) + 2*y2(t) + y3(t) + Derivative(y3(t), t) + exp(-2*t),0)] 
ics = {y1(0): 2, y2(0): 1, y3(0): 1} 
dsolve(ode,func=[y1(t),y2(t),y3(t)],ics=ics)
 

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