20.15.7 problem 7

Internal problem ID [3833]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 7
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 )&=\frac {x_{1} \left (t \right )}{t}+x_{2} \left (t \right ) t\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {x_{1} \left (t \right )}{t} \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 33

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

\begin{align*} \text {x1}(t)\to c_1 t \sin (t)-c_2 t \cos (t) \\ \text {x2}(t)\to c_1 \cos (t)+c_2 \sin (t) \\ \end{align*}