7.23.11 problem 11

Internal problem ID [597]
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 : 11
Date solved : Wednesday, February 05, 2025 at 03:45:57 AM
CAS classification : system_of_ODEs

\begin{align*} -\frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )-4 \frac {d}{d t}y \left (t \right )&=x \left (t \right )-15 y \left (t \right )+{\mathrm e}^{-t} \end{align*}

✓ Solution by Maple

Time used: 0.388 (sec). Leaf size: 63

dsolve([-diff(x(t),t)+2*diff(y(t),t)=x(t)+3*y(t)+exp(t),3*diff(x(t),t)-4*diff(y(t),t)=x(t)-15*y(t)+exp(-t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \sin \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_1 -\frac {{\mathrm e}^{-t}}{4}-\frac {11 \,{\mathrm e}^{t}}{20} \\ y \left (t \right ) &= \frac {\cos \left (3 t \right ) c_1}{3}-\frac {\cos \left (3 t \right ) c_2}{3}+\frac {\sin \left (3 t \right ) c_1}{3}+\frac {\sin \left (3 t \right ) c_2}{3}+\frac {{\mathrm e}^{t}}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.529 (sec). Leaf size: 77

DSolve[{-D[x[t],t]+2*D[y[t],t]==x[t]+3*y[t]+Exp[t],3*D[x[t],t]-4*D[y[t],t]==x[t]-15*y[t]+Exp[-t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {1}{20} e^{-t} \left (11 e^{2 t}+5\right )+c_1 \cos (3 t)+(c_1-3 c_2) \sin (3 t) \\ y(t)\to \frac {e^t}{10}+c_2 \cos (3 t)+\left (\frac {2 c_1}{3}-c_2\right ) \sin (3 t) \\ \end{align*}