50.27.4 problem 3(c)

Internal problem ID [8185]
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 10.2 Linear Systems. Page 380
Problem number : 3(c)
Date solved : Monday, January 27, 2025 at 03:45:32 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )+t -1\\ y^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right )-5 t -2 \end{align*}

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 43

dsolve([diff(x(t),t)=x(t)+2*y(t)+t-1,diff(y(t),t)=3*x(t)+2*y(t)-5*t-2],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-t}+{\mathrm e}^{4 t} c_{1} +3 t -2 \\ y &= -c_{2} {\mathrm e}^{-t}+\frac {3 \,{\mathrm e}^{4 t} c_{1}}{2}+3-2 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.232 (sec). Leaf size: 88

DSolve[{D[x[t],t]==x[t]+2*y[t]+t-1,D[y[t],t]==3*x[t]+2*y[t]-5*t-2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-t} \left (5 e^t (3 t-2)+2 (c_1+c_2) e^{5 t}+3 c_1-2 c_2\right ) \\ y(t)\to \frac {1}{5} e^{-t} \left (-5 e^t (2 t-3)+3 (c_1+c_2) e^{5 t}-3 c_1+2 c_2\right ) \\ \end{align*}