75.28.4 problem 790

Internal problem ID [17251]
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 21. Finding integrable combinations. Exercises page 219
Problem number : 790
Date solved : Tuesday, January 28, 2025 at 08:27:38 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.168 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 145

DSolve[{D[x[t],t]==y[t]/(x[t]-y[t]),D[y[t],t]==x[t]/(x[t]-y[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\frac {1}{2} \sqrt {\frac {\left (t^2-2 c_2 t+c_2{}^2+2 c_1\right ){}^2}{(t-c_2){}^2}} \\ x(t)\to -\frac {t^2-2 c_2 t+c_2{}^2-2 c_1}{2 t-2 c_2} \\ y(t)\to \frac {1}{2} \sqrt {\frac {\left (t^2-2 c_2 t+c_2{}^2+2 c_1\right ){}^2}{(t-c_2){}^2}} \\ x(t)\to -\frac {t^2-2 c_2 t+c_2{}^2-2 c_1}{2 t-2 c_2} \\ \end{align*}