Internal problem ID [15503]
Internal file name [OUTPUT/15504_Friday_May_10_2024_05_47_30_PM_95537058/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: 773.
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 {y \left (t \right )}{x \left (t \right )+y \left (t \right )}+\frac {t}{x \left (t \right )+y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{x \left (t \right )+y \left (t \right )}-\frac {t}{x \left (t \right )+y \left (t \right )} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 61
dsolve([diff(x(t),t)=(y(t)+t)/(x(t)+y(t)),diff(y(t),t)=(x(t)-t)/(x(t)+y(t))],singsol=all)
\begin{align*} \\ \left [\left \{x \left (t \right ) &= \frac {c_{1} t^{2}-c_{2} t +1}{c_{1} t -c_{2}}\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 \}\right ] \\ \end{align*}
✓ Solution by Mathematica
Time used: 67.434 (sec). Leaf size: 45
DSolve[{x'[t]==(y[t]+t)/(x[t]+y[t]),y'[t]==(x[t]-t)/(x[t]+y[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {t^2+c_1 t+c_2}{t+c_1} \\ y(t)\to \frac {c_1 t+c_1{}^2-c_2}{t+c_1} \\ \end{align*}