40.15.3 problem 12

Internal problem ID [6789]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 21. System of simultaneous linear equations. Supplemetary problems. Page 163
Problem number : 12
Date solved : Monday, January 27, 2025 at 02:30:34 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+7 y \left (t \right )&={\mathrm e}^{t}+2\\ -2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+3 y \left (t \right )&={\mathrm e}^{t}-1 \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 70

dsolve([diff(x(t),t)+x(t)+2*diff(y(t),t)+7*y(t)=exp(t)+2,-2*x(t)+diff(y(t),t)+3*y(t)=exp(t)-1],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_2 +{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +\frac {13}{17}-\frac {5 \,{\mathrm e}^{t}}{26} \\ y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_2 -{\mathrm e}^{-4 t} \cos \left (t \right ) c_1 +{\mathrm e}^{-4 t} \sin \left (t \right ) c_1 +\frac {2 \,{\mathrm e}^{t}}{13}+\frac {3}{17} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.197 (sec). Leaf size: 79

DSolve[{D[x[t],t]+x[t]+2*D[y[t],t]+7*y[t]==Exp[t]+2,-2*x[t]+D[y[t],t]+3*y[t]==Exp[t]-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {5 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)+\frac {13}{17} \\ y(t)\to \frac {2 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {3}{17} \\ \end{align*}