Internal problem ID [15513]
Internal file name [OUTPUT/15514_Friday_May_10_2024_05_47_32_PM_25585316/index.tex]
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: 783.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown" Unable to solve or complete the solution.
Unable to parse ODE.
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 48
dsolve([diff(x(t),t$2)=y(t),diff(y(t),t$2)=x(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}-c_{3} \sin \left (t \right )-c_{4} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} \sin \left (t \right )+c_{4} \cos \left (t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 172
DSolve[{x''[t]==y[t],y''[t]==x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (c_1 e^{2 t}+c_2 e^{2 t}+c_3 e^{2 t}+c_4 e^{2 t}+2 (c_1-c_3) e^t \cos (t)+2 (c_2-c_4) e^t \sin (t)+c_1-c_2+c_3-c_4\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (c_1 e^{2 t}+c_2 e^{2 t}+c_3 e^{2 t}+c_4 e^{2 t}-2 (c_1-c_3) e^t \cos (t)-2 (c_2-c_4) e^t \sin (t)+c_1-c_2+c_3-c_4\right ) \\ \end{align*}