75.30.1 problem 810

Internal problem ID [17263]
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. Methods of integrating nonhomogeneous linear systems with constant coefficients. Exercises page 234
Problem number : 810
Date solved : Tuesday, January 28, 2025 at 09:58:35 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+2 x \left (t \right )-y \left (t \right )&=-{\mathrm e}^{2 t}\\ \frac {d}{d t}y \left (t \right )+3 x \left (t \right )-2 y \left (t \right )&=6 \,{\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 42

dsolve([diff(x(t),t)+2*x(t)-y(t)=-exp(2*t),diff(y(t),t)+3*x(t)-2*y(t)=6*exp(2*t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 85

DSolve[{D[x[t],t]+2*x[t]-y[t]==-Exp[2*t],D[y[t],t]+3*x[t]-2*y[t]==6*Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (4 e^{3 t}+(c_2-c_1) e^{2 t}+3 c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (18 e^{3 t}-3 (c_1-c_2) e^{2 t}+3 c_1-c_2\right ) \\ \end{align*}