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: 4.234 (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)
\[ x \relax (t ) = -\frac {11 \,{\mathrm e}^{4 t} t}{10}-{\mathrm e}^{2 t} \sqrt {6}\, \sin \left (\sqrt {6}\, t \right ) c_{1}+{\mathrm e}^{2 t} \sqrt {6}\, \cos \left (\sqrt {6}\, t \right ) c_{2}-\frac {34 \,{\mathrm e}^{4 t}}{25}+{\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}+{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}-\frac {204 \cos \relax (t )}{97}-\frac {556 \sin \relax (t )}{97} \] \[ y \relax (t ) = {\mathrm e}^{2 t} \sin \left (\sqrt {6}\, t \right ) c_{2}+{\mathrm e}^{2 t} \cos \left (\sqrt {6}\, t \right ) c_{1}+\frac {3 \,{\mathrm e}^{4 t} t}{10}-\frac {11 \,{\mathrm e}^{4 t}}{50}-\frac {8 \cos \relax (t )}{97}-\frac {212 \sin \relax (t )}{97} \]
✓ Solution by Mathematica
Time used: 3.263 (sec). Leaf size: 150
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}{50} e^{4 t} (55 t+68)-\frac {4}{97} (139 \sin (t)+51 \cos (t))+\frac {1}{6} e^{2 t} \left (6 c_1 \cos \left (\sqrt {6} t\right )+\sqrt {6} (c_1-7 c_2) \sin \left (\sqrt {6} t\right )\right ) \\ y(t)\to \frac {1}{50} e^{4 t} (15 t-11)-\frac {4}{97} (53 \sin (t)+2 \cos (t))+\frac {1}{6} e^{2 t} \left (6 c_2 \cos \left (\sqrt {6} t\right )+\sqrt {6} (c_1-c_2) \sin \left (\sqrt {6} t\right )\right ) \\ \end{align*}