Internal problem ID [9500]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1925.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right ) y^{\prime }\left (t \right )+y^{\prime }\left (t \right ) t -y \left (t \right )&=0\\ x^{\prime }\left (t \right )^{2}+t x^{\prime }\left (t \right )+a y^{\prime }\left (t \right )-x \left (t \right )&=0 \end {align*}
✓ Solution by Maple
Time used: 3.078 (sec). Leaf size: 154
dsolve([diff(x(t),t)^2+t*diff(x(t),t)+a*diff(y(t),t)-x(t)=0,diff(x(t),t)*diff(y(t),t)+t*diff(y(t),t)-y(t)=0],[x(t), y(t)], singsol=all)
\begin{align*} \{y \left (t \right ) = 0\} \\ \left \{x \left (t \right ) = -\frac {t^{2}}{4}\right \} \\ \end{align*} \begin{align*} \{y \left (t \right ) = 0\} \\ \{x \left (t \right ) = c_{1}^{2}+c_{1} t\} \\ \end{align*} \begin{align*} \left \{y \left (t \right ) = -\frac {t^{3}}{27 a}\right \} \\ \left \{x \left (t \right ) = -\frac {t^{2}}{3}\right \} \\ \end{align*} \begin{align*} \left \{y \left (t \right ) = \frac {t^{2}}{4 c_{1}}+\frac {a t}{2 c_{1}^{2}}+\frac {a^{2}}{4 c_{1}^{3}}\right \} \\ \left \{x \left (t \right ) = \frac {-2 y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right ) t +3 y \left (t \right )^{2}}{\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\right \} \\ \end{align*} \begin{align*} \{y \left (t \right ) = c_{1} t +c_{2}\} \\ \left \{x \left (t \right ) = \frac {a \left (\frac {d}{d t}y \left (t \right )\right )^{3}-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right ) t +y \left (t \right )^{2}}{\left (\frac {d}{d t}y \left (t \right )\right )^{2}}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 27
DSolve[{x'[t]^2+t*x'[t]+a*y'[t]-x[t]==0,x'[t]*y'[t]+t*y'[t]-y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to a c_2+c_1 (t+c_1) \\ y(t)\to c_2 (t+c_1) \\ \end{align*}