90.33.6 problem 6

Internal problem ID [25480]
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 : 6
Date solved : Friday, October 03, 2025 at 12:01:59 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 36

ode:=[diff(y__1(t),t) = 1/2*y__1(t)-y__2(t)+5, diff(y__2(t),t) = -y__1(t)+1/2*y__2(t)-5]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} c_2 +{\mathrm e}^{\frac {3 t}{2}} c_1 -\frac {10}{3} \\ y_{2} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} c_2 -{\mathrm e}^{\frac {3 t}{2}} c_1 +\frac {10}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 80

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

\begin{align*} \text {y1}(t)&\to \frac {1}{6} \left (3 (c_1-c_2) e^{3 t/2}+3 (c_1+c_2) e^{-t/2}-20\right )\\ \text {y2}(t)&\to \frac {1}{6} \left (-3 (c_1-c_2) e^{3 t/2}+3 (c_1+c_2) e^{-t/2}+20\right ) \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 41

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

\[ \left [ y_{1}{\left (t \right )} = C_{1} e^{- \frac {t}{2}} - C_{2} e^{\frac {3 t}{2}} - \frac {10}{3}, \ y_{2}{\left (t \right )} = C_{1} e^{- \frac {t}{2}} + C_{2} e^{\frac {3 t}{2}} + \frac {10}{3}\right ] \]