73.27.1 problem 38.1

Internal problem ID [15757]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.1
Date solved : Tuesday, January 28, 2025 at 08:06:43 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 35

dsolve([diff(x(t),t)=2*y(t),diff(y(t),t)=1-2*x(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 112

DSolve[{D[x[t],t]==2*y[t],D[y[t],t]==1-2*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \cos (2 t) \int _1^t-\sin (2 K[1])dK[1]+\sin (2 t) \int _1^t\cos (2 K[2])dK[2]+c_1 \cos (2 t)+c_2 \sin (2 t) \\ y(t)\to -\sin (2 t) \int _1^t-\sin (2 K[1])dK[1]+\cos (2 t) \int _1^t\cos (2 K[2])dK[2]+c_2 \cos (2 t)-c_1 \sin (2 t) \\ \end{align*}