78.27.3 problem 6 (c)

Internal problem ID [18468]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 55. Linear systems. Problems at page 496
Problem number : 6 (c)
Date solved : Tuesday, January 28, 2025 at 11:50:30 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.020 (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 ) &= {\mathrm e}^{4 t} c_{2} +{\mathrm e}^{-t} c_{1} +3 t -2 \\ y &= \frac {3 \,{\mathrm e}^{4 t} c_{2}}{2}-{\mathrm e}^{-t} c_{1} +3-2 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.218 (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*}