54.10.23 problem 1938

Internal problem ID [13157]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1938
Date solved : Sunday, October 12, 2025 at 02:35:50 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 1.519 (sec). Leaf size: 1110

ode:=[(x(t)-y(t))*(x(t)-z(t))*diff(x(t),t) = f(t), (y(t)-x(t))*(y(t)-z(t))*diff(y(t),t) = f(t), (z(t)-x(t))*(z(t)-y(t))*diff(z(t),t) = f(t)]; 
dsolve(ode);
 

\begin{align*} \text {Expression too large to display} \\ \left \{y \left (t \right ) &= \frac {4 x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{3}-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right )-\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}, y \left (t \right ) = \frac {4 x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{3}-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right )+\sqrt {-16 \left (\frac {d}{d t}x \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2} f \left (t \right )^{2}}}{4 \left (\frac {d}{d t}x \left (t \right )\right )^{3}}\right \} \\ \left \{z \left (t \right ) &= \frac {x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right ) y \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )-f \left (t \right )}{x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )-y \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 1557

ode={(x[t]-y[t])*(x[t]-z[t])*D[x[t],t]==f[t],(y[t]-x[t])*(y[t]-z[t])*D[y[t],t]==f[t],(z[t]-x[t])*(z[t]-y[t])*D[z[t],t]==f[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
f = Function("f") 
ode=[Eq((x(t) - y(t))*(x(t) - z(t))*Derivative(x(t), t) - f(t),0),Eq((-x(t) + y(t))*(y(t) - z(t))*Derivative(y(t), t) - f(t),0),Eq((-x(t) + z(t))*(-y(t) + z(t))*Derivative(z(t), t) - f(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

NotImplementedError :