Internal problem ID [10262]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1930.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )^{2}+y \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )^{2}+z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.359 (sec). Leaf size: 314
dsolve([diff(x(t),t)=y(t)-z(t),diff(y(t),t)=x(t)^2+y(t),diff(z(t),t)=x(t)^2+z(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} \{z \left (t \right ) = c_{1} +c_{2} {\mathrm e}^{t}\} \{y \left (t \right ) = z \left (t \right )\} \left \{x \left (t \right ) = \sqrt {\frac {d}{d t}z \left (t \right )-z \left (t \right )}, x \left (t \right ) = -\sqrt {\frac {d}{d t}z \left (t \right )-z \left (t \right )}\right \} \end{align*} \begin{align*} \left \{z \left (t \right ) = -\frac {c_{3}^{2}}{4 c_{1}}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{t}+c_{3} t \,{\mathrm e}^{t}\right \} \left \{y \left (t \right ) = -\frac {-2 z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )+2 z \left (t \right )^{2}+\sqrt {\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} \left (\frac {d}{d t}z \left (t \right )\right )-\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} z \left (t \right )-2 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right )^{2}+2 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right ) z \left (t \right )+\left (\frac {d}{d t}z \left (t \right )\right )^{3}-\left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )}}{2 \left (\frac {d}{d t}z \left (t \right )-z \left (t \right )\right )}, y \left (t \right ) = \frac {2 z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )-2 z \left (t \right )^{2}+\sqrt {\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} \left (\frac {d}{d t}z \left (t \right )\right )-\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} z \left (t \right )-2 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right )^{2}+2 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right ) z \left (t \right )+\left (\frac {d}{d t}z \left (t \right )\right )^{3}-\left (\frac {d}{d t}z \left (t \right )\right )^{2} z \left (t \right )}}{-2 z \left (t \right )+2 \frac {d}{d t}z \left (t \right )}\right \} \left \{x \left (t \right ) = \frac {-\frac {d}{d t}z \left (t \right )+\frac {d^{2}}{d t^{2}}z \left (t \right )}{2 y \left (t \right )-2 z \left (t \right )}\right \} \end{align*}
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 127
DSolve[{x'[t]==y[t]-z[t],y'[t]==x[t]^2+y[t],z'[t]==x[t]^2+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{t-c_3}+c_1 y(t)\to e^{2 t-2 c_3}+(c_1+c_2) e^{t-c_3}+2 c_1 e^{t-c_3} \log \left (e^{t-c_3}\right )-c_1{}^2 z(t)\to e^{2 t-2 c_3}+(-1+c_1+c_2) e^{t-c_3}+2 c_1 e^{t-c_3} \log \left (e^{t-c_3}\right )-c_1{}^2 \end{align*}