Internal problem ID [5787]
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(b).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=-2 x \relax (t )+y \relax (t )-t +3\\ y^{\prime }\relax (t )&=x \relax (t )+4 y \relax (t )+t -2 \end {align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 90
dsolve([diff(x(t),t)=-2*x(t)+y(t)-t+3,diff(y(t),t)=x(t)+4*y(t)+t-2],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} \sqrt {10}-{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} \sqrt {10}-3 \,{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2}-3 \,{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1}-\frac {5 t}{9}+\frac {145}{81} \] \[ y \relax (t ) = {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2}+{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1}-\frac {t}{9}+\frac {2}{81} \]
✓ Solution by Mathematica
Time used: 1.418 (sec). Leaf size: 426
DSolve[{x'[t]==-2*x[t]+y[t]-t+3,y'[t]==x[t]+4*y[t]+t-2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{10} e^t \left (\sqrt {10} \sinh \left (\sqrt {10} t\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (5+\sqrt {10}\right ) K[2]+e^{2 \sqrt {10} K[2]} \left (-2 \left (-5+\sqrt {10}\right ) K[2]+3 \sqrt {10}-20\right )-3 \sqrt {10}-20\right )dK[2]+\left (10 \cosh \left (\sqrt {10} t\right )-3 \sqrt {10} \sinh \left (\sqrt {10} t\right )\right ) \int _1^t\left (\cosh \left (\sqrt {10} K[1]\right ) (K[1]-3) (\sinh (K[1])-\cosh (K[1]))+\frac {e^{-K[1]} (11-4 K[1]) \sinh \left (\sqrt {10} K[1]\right )}{\sqrt {10}}\right )dK[1]+10 c_1 \cosh \left (\sqrt {10} t\right )+\sqrt {10} (c_2-3 c_1) \sinh \left (\sqrt {10} t\right )\right ) \\ y(t)\to \frac {1}{10} e^t \left (\left (3 \sqrt {10} \sinh \left (\sqrt {10} t\right )+10 \cosh \left (\sqrt {10} t\right )\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (5+\sqrt {10}\right ) K[2]+e^{2 \sqrt {10} K[2]} \left (-2 \left (-5+\sqrt {10}\right ) K[2]+3 \sqrt {10}-20\right )-3 \sqrt {10}-20\right )dK[2]+\sqrt {10} \sinh \left (\sqrt {10} t\right ) \left (\int _1^t\left (\cosh \left (\sqrt {10} K[1]\right ) (K[1]-3) (\sinh (K[1])-\cosh (K[1]))+\frac {e^{-K[1]} (11-4 K[1]) \sinh \left (\sqrt {10} K[1]\right )}{\sqrt {10}}\right )dK[1]+c_1+3 c_2\right )+10 c_2 \cosh \left (\sqrt {10} t\right )\right ) \\ \end{align*}