76.8.4 problem 4

Internal problem ID [17493]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 10:39:29 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-\frac {5 y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=\frac {9 x \left (t \right )}{5}-y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 57

dsolve([diff(x(t),t)=2*x(t)-5/2*y(t),diff(y(t),t)=9/5*x(t)-y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_{1} +\sin \left (\frac {3 t}{2}\right ) c_{2} -\cos \left (\frac {3 t}{2}\right ) c_{1} +\cos \left (\frac {3 t}{2}\right ) c_{2} \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 84

DSolve[{D[x[t],t]==2*x[t]-5/2*y[t],D[y[t],t]==9/5*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{t/2} \left (3 c_1 \cos \left (\frac {3 t}{2}\right )+(3 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ y(t)\to \frac {1}{5} e^{t/2} \left (5 c_2 \cos \left (\frac {3 t}{2}\right )+(6 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ \end{align*}