50.29.13 problem 4(c)

Internal problem ID [8209]
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(c)
Date solved : Wednesday, March 05, 2025 at 05:32:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+y \left (t \right )-t +3\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-5 y \left (t \right )+t +1 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 105

ode:=[diff(x(t),t) = -4*x(t)+y(t)-t+3, diff(y(t),t) = -x(t)-5*y(t)+t+1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {4 t}{21}+\frac {39}{49} \\ y &= -\frac {{\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {1}{147}+\frac {5 t}{21} \\ \end{align*}

✓ Mathematica. Time used: 1.459 (sec). Leaf size: 131

ode={D[x[t],t]==-4*x[t]+y[t]-t+3,D[y[t],t]==-x[t]-5*y[t]+t+1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {4 t}{21}+c_1 e^{-9 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+\frac {(c_1+2 c_2) e^{-9 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}+\frac {39}{49} \\ y(t)\to \frac {5 t}{21}+c_2 e^{-9 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )-\frac {(2 c_1+c_2) e^{-9 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}-\frac {1}{147} \\ \end{align*}

✓ Sympy. Time used: 0.664 (sec). Leaf size: 224

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(t + 4*x(t) - y(t) + Derivative(x(t), t) - 3,0),Eq(-t + x(t) + 5*y(t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {4 t \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \frac {4 t \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {9 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {9 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + \frac {39 \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{49} + \frac {39 \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{49}, \ y{\left (t \right )} = C_{1} e^{- \frac {9 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{- \frac {9 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + \frac {5 t \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} + \frac {5 t \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \frac {\sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{147} - \frac {\cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{147}\right ] \]