75.32.4 problem 828

Internal problem ID [17281]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.3 dAlemberts method. Exercises page 243
Problem number : 828
Date solved : Tuesday, January 28, 2025 at 09:58:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+y \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )-{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 70

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

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