Internal problem ID [10252]
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: 0.031 (sec). Leaf size: 45
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)],singsol=all)
\begin{align*} \{x \left (t \right ) &= c_{2} +c_{3} {\mathrm e}^{t}\} \\ \{y \left (t \right ) &= \left (\int x \left (t \right )^{2} {\mathrm e}^{-t}d t +c_{1} \right ) {\mathrm e}^{t}\} \\ \{z \left (t \right ) &= -\frac {d}{d t}x \left (t \right )+y \left (t \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*}