10.19.5 problem 5

Internal problem ID [1446]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:57:27 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (\cos \left (t \right ) c_1 -2 c_2 \cos \left (t \right )-2 c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 54

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \end{align*}