60.10.17 problem 1930

Internal problem ID [11928]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1930
Date solved : Tuesday, January 28, 2025 at 06:24:09 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )^{2}+y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )^{2}+z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 44

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 &= -\frac {d}{d t}x \left (t \right )+y \left (t \right )\} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 109

DSolve[{D[x[t],t]==y[t]-z[t],D[y[t],t]==x[t]^2+y[t],D[z[t],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^{t-c_3} \left (\int _1^{c_1+e^{t-c_3}}\frac {K[1]^2}{(c_1-K[1]){}^2}dK[1]+c_2\right ) \\ z(t)\to e^{t-c_3} \left (\int _1^{c_1+e^{t-c_3}}\frac {K[1]^2}{(c_1-K[1]){}^2}dK[1]-1+c_2\right ) \\ \end{align*}