10.18.13 problem 13

Internal problem ID [1440]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 13
Date solved : Monday, January 27, 2025 at 04:57:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}+\frac {{\mathrm e}^{-\frac {t}{2}}}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=-1/2*x__1(t)-1/8*x__2(t)+1/2*exp(-t/2),diff(x__2(t),t)=2*x__1(t)-1/2*x__2(t)+0],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 69

DSolve[{D[ x1[t],t]==-1/2*x1[t]-1/8*x2[t]+1/2*Exp[-t/2],D[ x2[t],t]==2*x1[t]-1/2*x2[t]+0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t/2} \left (4 c_1 \cos \left (\frac {t}{2}\right )-c_2 \sin \left (\frac {t}{2}\right )\right ) \\ \text {x2}(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {t}{2}\right )+4 c_1 \sin \left (\frac {t}{2}\right )+4\right ) \\ \end{align*}