10.18.7 problem 7

Internal problem ID [1434]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:57:15 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 )+2 \,{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )-{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 43

dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)+2*exp(t),diff(x__2(t),t)=4*x__1(t)+1*x__2(t)-exp(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 80

DSolve[{D[ x1[t],t]==1*x1[t]+1*x2[t]+2*Exp[t],D[ x2[t],t]==4*x1[t]+1*x2[t]-Exp[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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