20.20.1 problem 1

Internal problem ID [3891]
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 : 1
Date solved : Tuesday, January 28, 2025 at 02:39:19 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 29

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

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