76.10.9 problem 9

Internal problem ID [17531]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 10:40:00 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 49

dsolve([diff(x(t),t)=2*x(t)-4*y(t),diff(y(t),t)=2*x(t)-2*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 48

DSolve[{D[x[t],t]==2*x[t]-4*y[t],D[y[t],t]==2*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 \cos (2 t)+(c_1-2 c_2) \sin (2 t) \\ y(t)\to c_2 \cos (2 t)+(c_1-c_2) \sin (2 t) \\ \end{align*}