Internal problem ID [5967]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&={\mathrm e}^{4 t} t +4 \sin \relax (t )-4 \,{\mathrm e}^{4 t}+3 x \relax (t )-7 y \relax (t )\\ y^{\prime }\relax (t )&=2 \,{\mathrm e}^{4 t} t +8 \sin \relax (t )+{\mathrm e}^{4 t}+x \relax (t )+y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 1.62 (sec). Leaf size: 131
dsolve([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)],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = -\frac {11 \,{\mathrm e}^{4 t} t}{10}+{\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}+{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}-\frac {34 \,{\mathrm e}^{4 t}}{25}-\frac {204 \cos \relax (t )}{97}-\frac {556 \sin \relax (t )}{97}+{\mathrm e}^{2 t} \sqrt {6}\, \cos \left (\sqrt {6}\, t \right ) c_{2}-{\mathrm e}^{2 t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right ) c_{1} \] \[ y \relax (t ) = {\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}+{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}+\frac {3 \,{\mathrm e}^{4 t} t}{10}-\frac {11 \,{\mathrm e}^{4 t}}{50}-\frac {8 \cos \relax (t )}{97}-\frac {212 \sin \relax (t )}{97} \]
✓ Solution by Mathematica
Time used: 1.22 (sec). Leaf size: 483
DSolve[{x'[t]==3*x[t]-7*y[t]+4*Sin[t]+(t-4)*Exp[4*t],y'[t]==x[t]+y[t]+8*Sin[t]+(2*t+1)*Exp[4*t]},{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]+\sqrt {6} \sin \left (\sqrt {6} t\right ) \left (-7 \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]+c_1-7 c_2\right )+6 c_1 \cos \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-c_2) \sin \left (\sqrt {6} t\right )\right ) \\ \end{align*}