29.12 problem 4(b)

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: 0.093 (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: 2.522 (sec). Leaf size: 188

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 {e^{t-\sqrt {10} t} \left (-100 e^{\left (\sqrt {10}-1\right ) t} (9 t-29)+81 \left (c_1 \left (\left (10-3 \sqrt {10}\right ) e^{2 \sqrt {10} t}+10+3 \sqrt {10}\right )+\sqrt {10} c_2 \left (e^{2 \sqrt {10} t}-1\right )\right )\right )}{1620} \\ y(t)\to \frac {e^{t-\sqrt {10} t} \left (-20 e^{\left (\sqrt {10}-1\right ) t} (9 t-2)+81 \left (\sqrt {10} c_1 \left (e^{2 \sqrt {10} t}-1\right )+c_2 \left (\left (10+3 \sqrt {10}\right ) e^{2 \sqrt {10} t}+10-3 \sqrt {10}\right )\right )\right )}{1620} \\ \end{align*}