46.9.10 problem 10

Internal problem ID [9674]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 06:28:50 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-7 y \left (t \right )+4 \sin \left (t \right )+\left (t -4\right ) {\mathrm e}^{4 t}\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right )+8 \sin \left (t \right )+\left (2 t +1\right ) {\mathrm e}^{4 t} \end{align*}

✓ Maple. Time used: 1.405 (sec). Leaf size: 133

ode:=[diff(x(t),t) = 3*x(t)-7*y(t)+4*sin(t)+(t-4)*exp(4*t), diff(y(t),t) = x(t)+y(t)+8*sin(t)+(2*t+1)*exp(4*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_2 +{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_1 -\frac {11 \,{\mathrm e}^{4 t} t}{10}-\frac {34 \,{\mathrm e}^{4 t}}{25}-\frac {204 \cos \left (t \right )}{97}-\frac {556 \sin \left (t \right )}{97} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{4 t} t}{10}-\frac {11 \,{\mathrm e}^{4 t}}{50}+\frac {{\mathrm e}^{2 t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right ) c_1}{7}-\frac {{\mathrm e}^{2 t} \sqrt {6}\, \cos \left (\sqrt {6}\, t \right ) c_2}{7}+\frac {{\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_2}{7}+\frac {{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_1}{7}-\frac {8 \cos \left (t \right )}{97}-\frac {212 \sin \left (t \right )}{97} \\ \end{align*}

✓ Mathematica. Time used: 2.636 (sec). Leaf size: 520

ode={D[x[t],t]==3*x[t]-7*y[t]+4*Sin[t]+(t-4)*Exp[4*t],D[y[t],t]==x[t]+y[t]+8*Sin[t]+(2*t+1)*Exp[4*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{2 t} \left (\left (\sqrt {6} \sin \left (\sqrt {6} t\right )+6 \cos \left (\sqrt {6} t\right )\right ) \int _1^t\frac {1}{6} e^{-2 K[1]} \left (6 \cos \left (\sqrt {6} K[1]\right ) \left (e^{4 K[1]} (K[1]-4)+4 \sin (K[1])\right )+\sqrt {6} \left (e^{4 K[1]} (13 K[1]+11)+52 \sin (K[1])\right ) \sin \left (\sqrt {6} K[1]\right )\right )dK[1]-7 \sqrt {6} \sin \left (\sqrt {6} t\right ) \int _1^t\frac {1}{6} e^{-2 K[2]} \left (6 \cos \left (\sqrt {6} K[2]\right ) \left (e^{4 K[2]} (2 K[2]+1)+8 \sin (K[2])\right )+\sqrt {6} \left (e^{4 K[2]} (K[2]+5)+4 \sin (K[2])\right ) \sin \left (\sqrt {6} K[2]\right )\right )dK[2]+6 c_1 \cos \left (\sqrt {6} t\right )+\sqrt {6} c_1 \sin \left (\sqrt {6} t\right )-7 \sqrt {6} c_2 \sin \left (\sqrt {6} t\right )\right )\\ y(t)&\to \frac {1}{6} e^{2 t} \left (\sqrt {6} \sin \left (\sqrt {6} t\right ) \int _1^t\frac {1}{6} e^{-2 K[1]} \left (6 \cos \left (\sqrt {6} K[1]\right ) \left (e^{4 K[1]} (K[1]-4)+4 \sin (K[1])\right )+\sqrt {6} \left (e^{4 K[1]} (13 K[1]+11)+52 \sin (K[1])\right ) \sin \left (\sqrt {6} K[1]\right )\right )dK[1]+\left (6 \cos \left (\sqrt {6} t\right )-\sqrt {6} \sin \left (\sqrt {6} t\right )\right ) \int _1^t\frac {1}{6} e^{-2 K[2]} \left (6 \cos \left (\sqrt {6} K[2]\right ) \left (e^{4 K[2]} (2 K[2]+1)+8 \sin (K[2])\right )+\sqrt {6} \left (e^{4 K[2]} (K[2]+5)+4 \sin (K[2])\right ) \sin \left (\sqrt {6} K[2]\right )\right )dK[2]+6 c_2 \cos \left (\sqrt {6} t\right )+\sqrt {6} c_1 \sin \left (\sqrt {6} t\right )-\sqrt {6} c_2 \sin \left (\sqrt {6} t\right )\right ) \end{align*}

✓ Sympy. Time used: 1.866 (sec). Leaf size: 372

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq((4 - t)*exp(4*t) - 3*x(t) + 7*y(t) - 4*sin(t) + Derivative(x(t), t),0),Eq((-2*t - 1)*exp(4*t) - x(t) - y(t) - 8*sin(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {11 t e^{4 t} \sin ^{2}{\left (\sqrt {6} t \right )}}{10} - \frac {11 t e^{4 t} \cos ^{2}{\left (\sqrt {6} t \right )}}{10} - \left (C_{1} + \sqrt {6} C_{2}\right ) e^{2 t} \sin {\left (\sqrt {6} t \right )} - \left (\sqrt {6} C_{1} - C_{2}\right ) e^{2 t} \cos {\left (\sqrt {6} t \right )} - \frac {34 e^{4 t} \sin ^{2}{\left (\sqrt {6} t \right )}}{25} - \frac {34 e^{4 t} \cos ^{2}{\left (\sqrt {6} t \right )}}{25} - \frac {556 \sin {\left (t \right )} \sin ^{2}{\left (\sqrt {6} t \right )}}{97} - \frac {556 \sin {\left (t \right )} \cos ^{2}{\left (\sqrt {6} t \right )}}{97} - \frac {204 \sin ^{2}{\left (\sqrt {6} t \right )} \cos {\left (t \right )}}{97} - \frac {204 \cos {\left (t \right )} \cos ^{2}{\left (\sqrt {6} t \right )}}{97}, \ y{\left (t \right )} = - C_{1} e^{2 t} \sin {\left (\sqrt {6} t \right )} + C_{2} e^{2 t} \cos {\left (\sqrt {6} t \right )} + \frac {3 t e^{4 t} \sin ^{2}{\left (\sqrt {6} t \right )}}{10} + \frac {3 t e^{4 t} \cos ^{2}{\left (\sqrt {6} t \right )}}{10} - \frac {11 e^{4 t} \sin ^{2}{\left (\sqrt {6} t \right )}}{50} - \frac {11 e^{4 t} \cos ^{2}{\left (\sqrt {6} t \right )}}{50} - \frac {212 \sin {\left (t \right )} \sin ^{2}{\left (\sqrt {6} t \right )}}{97} - \frac {212 \sin {\left (t \right )} \cos ^{2}{\left (\sqrt {6} t \right )}}{97} - \frac {8 \sin ^{2}{\left (\sqrt {6} t \right )} \cos {\left (t \right )}}{97} - \frac {8 \cos {\left (t \right )} \cos ^{2}{\left (\sqrt {6} t \right )}}{97}\right ] \]