7.23.9 problem 9

Internal problem ID [595]
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 : 9
Date solved : Monday, January 27, 2025 at 02:55:08 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.110 (sec). Leaf size: 54

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

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+{\mathrm e}^{t} c_1 -\frac {7 \cos \left (2 t \right )}{5}-\frac {4 \sin \left (2 t \right )}{5} \\ y \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{t} c_1}{3}-\frac {4 \sin \left (2 t \right )}{5}-\frac {2 \cos \left (2 t \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 102

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

\begin{align*} x(t)\to \frac {1}{10} \left (-8 \sin (2 t)-14 \cos (2 t)+5 e^{-t} \left (c_1 \left (3 e^{2 t}-1\right )-3 c_2 \left (e^{2 t}-1\right )\right )\right ) \\ y(t)\to \frac {1}{10} \left (-8 \sin (2 t)-4 \cos (2 t)+e^{-t} \left (5 c_1 \left (e^{2 t}-1\right )-5 c_2 \left (e^{2 t}-3\right )\right )\right ) \\ \end{align*}