10.16.4 problem 4

Internal problem ID [1404]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 4
Date solved : Monday, January 27, 2025 at 04:56:49 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {9 x_{1} \left (t \right )}{5}-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 57

dsolve([diff(x__1(t),t)=2*x__1(t)-5/2*x__2(t),diff(x__2(t),t)=9/5*x__1(t)-1*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \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 ) \\ x_{2} \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: 0.005 (sec). Leaf size: 84

DSolve[{D[ x1[t],t]==2*x1[t]-5/2*x2[t],D[ x2[t],t]==9/5*x1[t]-1*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(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 ) \\ \text {x2}(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*}