82.55.1 problem Ex. 1

Internal problem ID [19036]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. problems at page 129
Problem number : Ex. 1
Date solved : Tuesday, January 28, 2025 at 12:46:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 x \left (t \right )+y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+5 x \left (t \right )+3 y \left (t \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 37

dsolve([diff(x(t),t)+diff(y(t),t)+2*x(t)+y(t)=0,diff(y(t),t)+5*x(t)+3*y(t)=0],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 42

DSolve[{D[x[t],t]+D[y[t],t]+2*x[t]+y[t]==0,D[y[t],t]+5*x[t]+3*y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 2 c_2 \sin (t)+c_1 (3 \sin (t)+\cos (t)) \\ y(t)\to c_2 (\cos (t)-3 \sin (t))-5 c_1 \sin (t) \\ \end{align*}