Internal problem ID [9504]
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 }\relax (t ) y^{\prime }\relax (t )+y^{\prime }\relax (t ) t -y \relax (t )&=0\\ x^{\prime }\relax (t )^{2}+t x^{\prime }\relax (t )+a y^{\prime }\relax (t )-x \relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 220
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*} \left \{x \relax (t ) = -\frac {t^{2}}{3}\right \} \\ \left \{y \relax (t ) = -\frac {t^{3}}{27 a}\right \} \\ \end{align*} \begin{align*} \{x \relax (t ) = t c_{1}+c_{2}\} \\ \left \{y \relax (t ) = \frac {-\left (\frac {d}{d t}x \relax (t )\right )^{3}-2 \left (\frac {d}{d t}x \relax (t )\right )^{2} t -t^{2} \left (\frac {d}{d t}x \relax (t )\right )+x \relax (t ) \left (\frac {d}{d t}x \relax (t )\right )+x \relax (t ) t}{a}\right \} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = -\frac {5 t^{2}}{12}-\frac {t \left (-t -\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4}, x \relax (t ) = -\frac {5 t^{2}}{12}-\frac {t \left (-t +\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4}, x \relax (t ) = -\frac {5 t^{2}}{12}+\frac {t \left (t -\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4}, x \relax (t ) = -\frac {5 t^{2}}{12}+\frac {t \left (t +\sqrt {3}\, c_{1}\right )}{6}+\frac {c_{1}^{2}}{4}\right \} \\ \left \{y \relax (t ) = -\frac {-2 t^{2} \left (\frac {d}{d t}x \relax (t )\right )-2 t^{3}-6 x \relax (t ) \left (\frac {d}{d t}x \relax (t )\right )-7 x \relax (t ) t}{9 a}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.009 (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*}