10.16.11 problem 11

Internal problem ID [1411]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 11
Date solved : Monday, January 27, 2025 at 04:56:56 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=\frac {3 x_{1} \left (t \right )}{4}-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{4} \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 47

dsolve([diff(x__1(t),t)=3/4*x__1(t)-2*x__2(t),diff(x__2(t),t)=1*x__1(t)-5/4*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) c_1 -c_2 \cos \left (t \right )-c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 56

DSolve[{D[ x1[t],t]==3/4*x1[t]-2*x2[t],D[ x2[t],t]==1*x1[t]-5/4*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)+(c_1-c_2) \sin (t)) \\ \end{align*}