Internal problem ID [15504]
Internal file name [OUTPUT/15505_Friday_May_10_2024_05_47_30_PM_48338425/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 19. Basic concepts and definitions.
Exercises page 199
Problem number: 774.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {t}{y \left (t \right )-x \left (t \right )}-\frac {y \left (t \right )}{y \left (t \right )-x \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{y \left (t \right )-x \left (t \right )}-\frac {t}{y \left (t \right )-x \left (t \right )} \end {align*}
✓ Solution by Maple
Time used: 0.422 (sec). Leaf size: 132
dsolve([diff(x(t),t)=(t-y(t))/(y(t)-x(t)),diff(y(t),t)=(x(t)-t)/(y(t)-x(t))],singsol=all)
\begin{align*} \left \{x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{-4+3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}-\sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_{1}}{2}} \textit {\_f}}d \textit {\_f} +c_{2} \right ), x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_{1}} \textit {\_f}^{2}-1\right )}{3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+\sqrt {-3 \,{\mathrm e}^{c_{1}} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_{1}}{2}} \textit {\_f} -4}d \textit {\_f} +c_{2} \right )\right \} \\ \left \{y \left (t \right ) &= \frac {x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+t}{\frac {d}{d t}x \left (t \right )+1}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.351 (sec). Leaf size: 151
DSolve[{x'[t]==(t-y[t])/(y[t]-x[t]),y'[t]==(x[t]-t)/(y[t]-x[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ y(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ x(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ y(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ \end{align*}