44.28.9 problem 5(b)

Internal problem ID [9482]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section 10.3 Homogeneous Linear Systems with Constant Coefficients. Page 387
Problem number : 5(b)
Date solved : Tuesday, September 30, 2025 at 06:19:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )-5 t +2\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-2 y \left (t \right )-8 t -8 \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 42

ode:=[diff(x(t),t) = x(t)+y(t)-5*t+2, diff(y(t),t) = 4*x(t)-2*y(t)-8*t-8]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} c_2 +{\mathrm e}^{-3 t} c_1 +3 t +2 \\ y \left (t \right ) &= {\mathrm e}^{2 t} c_2 -4 \,{\mathrm e}^{-3 t} c_1 -1+2 t \\ \end{align*}

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 251

ode={D[x[t],t]==x[t]+y[t]-5*t+2,D[y[t],t]==4*x[t]-2*y[t]-8*t-8}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} e^{-3 t} \left (\left (4 e^{5 t}+1\right ) \int _1^t\frac {1}{5} e^{-2 K[1]} \left (e^{5 K[1]} (3 K[1]+10)-28 K[1]\right )dK[1]+\left (e^{5 t}-1\right ) \int _1^t-\frac {4}{5} e^{-2 K[2]} \left (7 K[2]+e^{5 K[2]} (3 K[2]+10)\right )dK[2]+4 c_1 e^{5 t}+c_2 e^{5 t}+c_1-c_2\right )\\ y(t)&\to \frac {1}{5} e^{-3 t} \left (4 \left (e^{5 t}-1\right ) \int _1^t\frac {1}{5} e^{-2 K[1]} \left (e^{5 K[1]} (3 K[1]+10)-28 K[1]\right )dK[1]+\left (e^{5 t}+4\right ) \int _1^t-\frac {4}{5} e^{-2 K[2]} \left (7 K[2]+e^{5 K[2]} (3 K[2]+10)\right )dK[2]+4 c_1 e^{5 t}+c_2 e^{5 t}-4 c_1+4 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*t - x(t) - y(t) + Derivative(x(t), t) - 2,0),Eq(8*t - 4*x(t) + 2*y(t) + Derivative(y(t), t) + 8,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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