20.20.27 problem 27

Internal problem ID [3917]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 27
Date solved : Monday, January 27, 2025 at 08:04:29 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=10 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+\frac {{\mathrm e}^{6 t}}{t} \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 56

dsolve([diff(x__1(t),t)=10*x__1(t)-4*x__2(t),diff(x__2(t),t)=4*x__1(t)+2*x__2(t)+1/t*exp(6*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{6 t} \left (4 t \ln \left (t \right )-c_{1} t -c_{2} -4 t \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{6 t} \left (16 t \ln \left (t \right )-4 c_{1} t -4 \ln \left (t \right )+c_{1} -4 c_{2} -16 t \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 61

DSolve[{D[x1[t],t]==10*x1[t]-4*x2[t],D[x2[t],t]==4*x1[t]+2*x2[t]+1/t*Exp[6*t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{6 t} (-4 t \log (t)+4 (1+c_1-c_2) t+c_1) \\ \text {x2}(t)\to e^{6 t} ((1-4 t) \log (t)+4 (1+c_1-c_2) t+c_2) \\ \end{align*}