75.27.1 problem 776

Internal problem ID [17237]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 776
Date solved : Tuesday, January 28, 2025 at 09:58:19 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-9 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 35

dsolve([diff(x(t),t)=-9*y(t),diff(y(t),t)=x(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 42

DSolve[{D[x[t],t]==-9*y[t],D[y[t],t]==x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 \cos (3 t)-3 c_2 \sin (3 t) \\ y(t)\to c_2 \cos (3 t)+\frac {1}{3} c_1 \sin (3 t) \\ \end{align*}