7.23.7 problem 7

Internal problem ID [593]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 7
Date solved : Monday, January 27, 2025 at 02:55:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 43

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

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{2 t}+c_1 \,{\mathrm e}^{3 t}-\frac {t}{3}+\frac {1}{18} \\ y \left (t \right ) &= -2 c_2 \,{\mathrm e}^{2 t}-c_1 \,{\mathrm e}^{3 t}-\frac {5}{9}-\frac {2 t}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.102 (sec). Leaf size: 77

DSolve[{D[x[t],t]==4*x[t]+y[t]+2*t,D[y[t],t]==-2*x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {t}{3}-(c_1+c_2) e^{2 t}+(2 c_1+c_2) e^{3 t}+\frac {1}{18} \\ y(t)\to -\frac {2 t}{3}+2 (c_1+c_2) e^{2 t}-(2 c_1+c_2) e^{3 t}-\frac {5}{9} \\ \end{align*}