75.26.6 problem 773

Internal problem ID [17234]
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
Date solved : Tuesday, January 28, 2025 at 08:27:34 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.188 (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: 63.812 (sec). Leaf size: 45

DSolve[{D[x[t],t]==(y[t]+t)/(x[t]+y[t]),D[y[t],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*}