44.29.11 problem 4(a)

Internal problem ID [9493]
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 A. Drill exercises. Page 400
Problem number : 4(a)
Date solved : Tuesday, September 30, 2025 at 06:19:34 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 )-4 t +1\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+2 y \left (t \right )+3 t +4 \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 105

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2 +{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1 +\frac {7 t}{2}+\frac {25}{8} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}-\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}-\frac {5}{16}+\frac {t}{4} \\ \end{align*}

✓ Mathematica. Time used: 1.086 (sec). Leaf size: 494

ode={D[x[t],t]==x[t]+2*y[t]-4+t+1,D[y[t],t]==-x[t]+2*y[t]+3*t+4}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{7} e^{3 t/2} \left (\left (7 \cos \left (\frac {\sqrt {7} t}{2}\right )-\sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) (K[1]-3)-\sqrt {7} (11 K[1]+19) \sin \left (\frac {1}{2} \sqrt {7} K[1]\right )\right )dK[1]+4 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[2]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right ) (3 K[2]+4)-\sqrt {7} (K[2]+10) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]+7 c_1 \cos \left (\frac {\sqrt {7} t}{2}\right )-\sqrt {7} c_1 \sin \left (\frac {\sqrt {7} t}{2}\right )+4 \sqrt {7} c_2 \sin \left (\frac {\sqrt {7} t}{2}\right )\right )\\ y(t)&\to \frac {1}{7} e^{3 t/2} \left (-2 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) (K[1]-3)-\sqrt {7} (11 K[1]+19) \sin \left (\frac {1}{2} \sqrt {7} K[1]\right )\right )dK[1]+\left (\sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )+7 \cos \left (\frac {\sqrt {7} t}{2}\right )\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[2]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right ) (3 K[2]+4)-\sqrt {7} (K[2]+10) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]+7 c_2 \cos \left (\frac {\sqrt {7} t}{2}\right )-2 \sqrt {7} c_1 \sin \left (\frac {\sqrt {7} t}{2}\right )+\sqrt {7} c_2 \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.456 (sec). Leaf size: 224

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*t - x(t) - 2*y(t) + Derivative(x(t), t) - 1,0),Eq(-3*t + x(t) - 2*y(t) + Derivative(y(t), t) - 4,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {7 t \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} + \frac {7 t \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} - \left (\frac {C_{1}}{2} - \frac {\sqrt {7} C_{2}}{2}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + \left (\frac {\sqrt {7} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {25 \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{8} + \frac {25 \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{8}, \ y{\left (t \right )} = - C_{1} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{2} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {t \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} + \frac {t \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} - \frac {5 \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{16} - \frac {5 \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{16}\right ] \]