7.23.8 problem 8

Internal problem ID [594]
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 : 8
Date solved : Monday, January 27, 2025 at 02:55:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 66

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

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