90.34.13 problem 13

Internal problem ID [25497]
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 : 13
Date solved : Friday, October 03, 2025 at 12:02:09 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 46

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

\begin{align*} \text {y1}(t)&\to 2 e^{-t} (\cos (2 t)+4 \sin (t) \cos (t))\\ \text {y2}(t)&\to e^{-t} \left (e^t+\sin (2 t)-2 \cos (2 t)\right ) \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 56

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

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