50.29.12 problem 4(b)

Internal problem ID [8208]
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)
Date solved : Monday, January 27, 2025 at 03:46:01 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y \left (t \right )-t +3\\ y^{\prime }\left (t \right )&=x \left (t \right )+4 y \left (t \right )+t -2 \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 89

dsolve([diff(x(t),t)=-2*x(t)+y(t)-t+3,diff(y(t),t)=x(t)+4*y(t)+t-2],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_{2} +{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_{1} -\frac {5 t}{9}+\frac {145}{81} \\ y &= {\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 {t}{9}+\frac {2}{81} \\ \end{align*}

✓ Solution by Mathematica

Time used: 11.075 (sec). Leaf size: 190

DSolve[{D[x[t],t]==-2*x[t]+y[t]-t+3,D[y[t],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 (\left (3 \sqrt {10}-10\right ) c_1-\sqrt {10} c_2\right ) e^{2 \sqrt {10} t}-81 \left (10+3 \sqrt {10}\right ) c_1+81 \sqrt {10} c_2\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 (10+3 \sqrt {10}\right ) c_2\right ) e^{2 \sqrt {10} t}-81 \left (\sqrt {10} c_1+\left (3 \sqrt {10}-10\right ) c_2\right )\right )}{1620} \\ \end{align*}