75.28.7 problem 793

Internal problem ID [17183]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 21. Finding integrable combinations. Exercises page 219
Problem number : 793
Date solved : Monday, March 31, 2025 at 03:44:02 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}

✗ Maple

ode:=[diff(x(t),t) = cos(x(t))^2*cos(y(t))^2+sin(x(t))^2*cos(y(t))^2, diff(y(t),t) = -1/2*sin(2*x(t))*sin(2*y(t))]; 
ic:=x(0) = 0y(0) = 0; 
dsolve([ode,ic]);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],t]==Cos[x[t]]^2*Cos[y[t]]^2+Sin[x[t]]^2*Cos[y[t]]^2,D[y[t],t]==-1/2*Sin[2*x[t]]*Sin[2*y[t]]}; 
ic={x[0]==0,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

{}

✗ Sympy

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

Timed Out