Internal problem ID [9503]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1924.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} \left (x \relax (t )^{2}+y \relax (t )^{2}-t^{2}\right ) x^{\prime }\relax (t )&=-2 x \relax (t ) t\\ \left (x \relax (t )^{2}+y \relax (t )^{2}-t^{2}\right ) y^{\prime }\relax (t )&=-2 t y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.499 (sec). Leaf size: 186
dsolve({(x(t)^2+y(t)^2-t^2)*diff(x(t),t)=-2*t*x(t),(x(t)^2+y(t)^2-t^2)*diff(y(t),t)=-2*t*y(t)},{x(t), y(t)}, singsol=all)
\begin{align*} \{x \relax (t ) = 0\} \\ \left \{y \relax (t ) = \frac {1+\sqrt {-4 c_{1}^{2} t^{2}+1}}{2 c_{1}}, y \relax (t ) = -\frac {-1+\sqrt {-4 c_{1}^{2} t^{2}+1}}{2 c_{1}}\right \} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = -\frac {-c_{1}+\sqrt {-2 c_{2} t^{2}+c_{1}^{2}}}{2 c_{2}}, x \relax (t ) = \frac {c_{1}+\sqrt {-2 c_{2} t^{2}+c_{1}^{2}}}{2 c_{2}}\right \} \\ \left \{y \relax (t ) = \frac {\sqrt {-\left (\frac {d}{d t}x \relax (t )\right ) \left (x \relax (t )^{2} \left (\frac {d}{d t}x \relax (t )\right )-t^{2} \left (\frac {d}{d t}x \relax (t )\right )+2 x \relax (t ) t \right )}}{\frac {d}{d t}x \relax (t )}, y \relax (t ) = -\frac {\sqrt {-\left (\frac {d}{d t}x \relax (t )\right ) \left (x \relax (t )^{2} \left (\frac {d}{d t}x \relax (t )\right )-t^{2} \left (\frac {d}{d t}x \relax (t )\right )+2 x \relax (t ) t \right )}}{\frac {d}{d t}x \relax (t )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 179
DSolve[{(x[t]^2+y[t]^2-t^2)*x'[t]==-2*t*x[t],(x[t]^2+y[t]^2-t^2)*y'[t]==-2*t*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {c_1 \left (e^{c_2}-\sqrt {e^{2 c_2}-4 \left (1+c_1{}^2\right ) t^2}\right )}{2 \left (1+c_1{}^2\right )} \\ x(t)\to \frac {e^{c_2}-\sqrt {e^{2 c_2}-4 \left (1+c_1{}^2\right ) t^2}}{2 \left (1+c_1{}^2\right )} \\ y(t)\to \frac {c_1 \left (\sqrt {e^{2 c_2}-4 \left (1+c_1{}^2\right ) t^2}+e^{c_2}\right )}{2 \left (1+c_1{}^2\right )} \\ x(t)\to \frac {\sqrt {e^{2 c_2}-4 \left (1+c_1{}^2\right ) t^2}+e^{c_2}}{2 \left (1+c_1{}^2\right )} \\ \end{align*}