10.18.10 problem 10

Internal problem ID [1437]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 10
Date solved : Monday, January 27, 2025 at 04:57:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+\sqrt {2}\, x_{2} \left (t \right )+{\mathrm e}^{-t}\\ \frac {d}{d t}x_{2} \left (t \right )&=\sqrt {2}\, x_{1} \left (t \right )-2 x_{2} \left (t \right )-{\mathrm e}^{-t} \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 89

dsolve([diff(x__1(t),t)=-3*x__1(t)+sqrt(2)*x__2(t)+exp(-t),diff(x__2(t),t)=sqrt(2)*x__1(t)-2*x__2(t)-exp(-t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{-4 t}+{\mathrm e}^{-t} c_1 -\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}+\frac {t \,{\mathrm e}^{-t}}{3} \\ x_{2} \left (t \right ) &= -\frac {2 t \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-4 t} \sqrt {2}\, c_2}{2}-\frac {{\mathrm e}^{-t}}{3}+\sqrt {2}\, {\mathrm e}^{-t} c_1 +\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}-\frac {\sqrt {2}\, {\mathrm e}^{-t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 128

DSolve[{D[ x1[t],t]==-3*x1[t]+Sqrt[2]*x2[t]+Exp[-t],D[ x2[t],t]==Sqrt[2]*x1[t]-2*x2[t]-Exp[-t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (-3 \left (\sqrt {2}-1\right ) t+\sqrt {2}+2+3 c_1+3 \sqrt {2} c_2\right )+6 c_1-3 \sqrt {2} c_2\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (3 \left (\sqrt {2}-2\right ) t-\sqrt {2}-1+3 \sqrt {2} c_1+6 c_2\right )-3 \sqrt {2} c_1+3 c_2\right ) \\ \end{align*}