10.19.15 problem 15

Internal problem ID [1456]
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 : 15
Date solved : Monday, January 27, 2025 at 04:57:35 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 60

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

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

✓ Solution by Mathematica

Time used: 0.344 (sec). Leaf size: 85

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

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