44.27.4 problem 3(c)

Internal problem ID [9471]
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 10.2 Linear Systems. Page 380
Problem number : 3(c)
Date solved : Tuesday, September 30, 2025 at 06:19:13 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+t -1\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+2 y \left (t \right )-5 t -2 \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 43

ode:=[diff(x(t),t) = x(t)+2*y(t)+t-1, diff(y(t),t) = 3*x(t)+2*y(t)-5*t-2]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.143 (sec). Leaf size: 262

ode={D[x[t],t]==x[t]+2*y[t]+t-1,D[y[t],t]==3*x[t]+2*y[t]-5*t-2}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} e^{-t} \left (\left (2 e^{5 t}+3\right ) \int _1^t\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]+2 \left (e^{5 t}-1\right ) \int _1^t-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+2 c_1 e^{5 t}+2 c_2 e^{5 t}+3 c_1-2 c_2\right )\\ y(t)&\to \frac {1}{5} e^{-t} \left (3 \left (e^{5 t}-1\right ) \int _1^t\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]+\left (3 e^{5 t}+2\right ) \int _1^t-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+3 c_1 e^{5 t}+3 c_2 e^{5 t}-3 c_1+2 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t - x(t) - 2*y(t) + Derivative(x(t), t) + 1,0),Eq(5*t - 3*x(t) - 2*y(t) + Derivative(y(t), t) + 2,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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