Internal problem ID [9509]
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 }\relax (t )&=y \relax (t )-z \relax (t )\\ y^{\prime }\relax (t )&=x \relax (t )^{2}+y \relax (t )\\ z^{\prime }\relax (t )&=x \relax (t )^{2}+z \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.056 (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)},{x(t), y(t), z(t)}, singsol=all)
\begin{align*} x \relax (t ) = c_{2}+c_{3} {\mathrm e}^{t} \\ \end{align*} \begin{align*} y \relax (t ) = \left (\int x \relax (t )^{2} {\mathrm e}^{-t}d t +c_{1}\right ) {\mathrm e}^{t} \\ \end{align*} \begin{align*} z \relax (t ) = -\frac {d}{d t}x \relax (t )+y \relax (t ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 111
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}+e^{t-c_3} \left (2 c_1 \log \left (e^{t-c_3}\right )+c_1+c_2\right )-c_1{}^2 \\ z(t)\to e^{2 t-2 c_3}+e^{t-c_3} \left (2 c_1 \log \left (e^{t-c_3}\right )-1+c_1+c_2\right )-c_1{}^2 \\ \end{align*}