75.28.6 problem 792

Internal problem ID [17253]
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 : 792
Date solved : Tuesday, January 28, 2025 at 08:27:39 PM
CAS classification : system_of_ODEs

\begin{align*} {\mathrm e}^{t} \left (\frac {d}{d t}x \left (t \right )\right )&=\frac {1}{y \left (t \right )}\\ {\mathrm e}^{t} \left (\frac {d}{d t}y \left (t \right )\right )&=\frac {1}{x \left (t \right )} \end{align*}

✓ Solution by Maple

Time used: 0.148 (sec). Leaf size: 51

dsolve([exp(t)*diff(x(t),t)=1/y(t),exp(t)*diff(y(t),t)=1/x(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 125

DSolve[{Exp[t]*D[x[t],t]==1/y[t],Exp[t]*D[y[t],t]==1/x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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