66.3.4 problem Problem 5

Internal problem ID [13876]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 3, SYSTEMS OF DIFFERENTIAL EQUATIONS. Problems page 209
Problem number : Problem 5
Date solved : Monday, March 31, 2025 at 08:15:56 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.171 (sec). Leaf size: 19

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = y(t)^2/x(t)]; 
dsolve(ode);
 

\begin{align*} \{x \left (t \right ) &= {\mathrm e}^{c_1 t} c_2\} \\ \{y \left (t \right ) &= \frac {d}{d t}x \left (t \right )\} \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 28

ode={D[x[t],t]==y[t],D[y[t],t]==y[t]^2/x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to c_1 c_2 e^{c_1 t} \\ x(t)\to c_2 e^{c_1 t} \\ \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(Derivative(y(t), t) - y(t)**2/x(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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