76.6.15 problem 17

Internal problem ID [17399]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 17
Date solved : Monday, March 31, 2025 at 04:12:52 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {x \left (t \right )}{4}-\frac {3 y \left (t \right )}{4}+8\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{2}+y \left (t \right )-\frac {23}{2} \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 38

ode:=[diff(x(t),t) = -1/4*x(t)-3/4*y(t)+8, diff(y(t),t) = 1/2*x(t)+y(t)-23/2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{\frac {t}{2}} c_1 +{\mathrm e}^{\frac {t}{4}} c_2 +5 \\ y \left (t \right ) &= -4 \,{\mathrm e}^{\frac {t}{2}} c_1 -\frac {2 \,{\mathrm e}^{\frac {t}{4}} c_2}{3}+9 \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 75

ode={D[x[t],t]==-1/4*x[t]-75/100*y[t]+8,D[y[t],t]==1/2*x[t]+y[t]-115/10}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 3 (c_1+c_2) e^{t/4}-(2 c_1+3 c_2) e^{t/2}+5 \\ y(t)\to -2 (c_1+c_2) e^{t/4}+(2 c_1+3 c_2) e^{t/2}+9 \\ \end{align*}

✓ Sympy. Time used: 0.177 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t)/4 + 3*y(t)/4 + Derivative(x(t), t) - 8,0),Eq(-x(t)/2 - y(t) + Derivative(y(t), t) + 23/2,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {3 C_{1} e^{\frac {t}{4}}}{2} - C_{2} e^{\frac {t}{2}} + 5, \ y{\left (t \right )} = C_{1} e^{\frac {t}{4}} + C_{2} e^{\frac {t}{2}} + 9\right ] \]