54.10.12 problem 1926

Internal problem ID [13146]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1926
Date solved : Sunday, October 12, 2025 at 02:35:48 AM
CAS classification : system_of_ODEs

\begin{align*} x \left (t \right )&=t \left (\frac {d}{d t}x \left (t \right )\right )+f \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right )\\ y \left (t \right )&=t \left (\frac {d}{d t}y \left (t \right )\right )+g \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.181 (sec). Leaf size: 95

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

\begin{align*} \{\int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_1 &= t \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )+f \left (\operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right ), \frac {d}{d t}y \left (t \right )\right )\} \\ \{x \left (t \right ) &= \int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_1\} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 28

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

\begin{align*} x(t)&\to f(c_1,c_2)+c_1 t\\ y(t)&\to g(c_1,c_2)+c_2 t \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 17

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

\[ \left \{x{\left (t \right )} = C_{1} t + f{\left (C_{1},C_{2} \right )}, y{\left (t \right )} = C_{2} t + g{\left (C_{1},C_{2} \right )}\right \} \]