64.24.6 problem 6

Internal problem ID [13691]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Nonlinear differential equations. Section 13.2, Exercises page 656
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 05:55:19 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 57

dsolve([diff(x(t),t)=x(t)-2*y(t),diff(y(t),t)=4*x(t)+5*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{3 t} \left (\cos \left (2 t \right ) c_{1} +c_{2} \cos \left (2 t \right )+c_{1} \sin \left (2 t \right )-\sin \left (2 t \right ) c_{2} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 59

DSolve[{D[x[t],t]==x[t]-2*y[t],D[y[t],t]==4*x[t]+5*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{3 t} (c_1 \cos (2 t)-(c_1+c_2) \sin (2 t)) \\ y(t)\to e^{3 t} (c_2 \cos (2 t)+(2 c_1+c_2) \sin (2 t)) \\ \end{align*}