15.17.4 problem 4

Internal problem ID [3240]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 26, page 115
Problem number : 4
Date solved : Monday, January 27, 2025 at 07:27:18 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 2.125 (sec). Leaf size: 41

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

\begin{align*} x \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{5 t}}{20}-\frac {16 \cos \left (t \right )}{13}-\frac {2 \sin \left (t \right )}{13}+c_2 \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{5 t}}{5}+\frac {8 \sin \left (t \right )}{13}-\frac {14 \cos \left (t \right )}{13}+c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 80

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

\begin{align*} x(t)\to \frac {1}{65} \left (-10 \sin (t)-80 \cos (t)+13 c_1 \left (e^{5 t}+4\right )-13 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{65} \left (40 \sin (t)-70 \cos (t)-52 c_1 \left (e^{5 t}-1\right )+13 c_2 \left (4 e^{5 t}+1\right )\right ) \\ \end{align*}