60.10.13 problem 1927

Internal problem ID [11848]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1927
Date solved : Sunday, March 30, 2025 at 09:18:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=a \,{\mathrm e}^{2 x \left (t \right )}-{\mathrm e}^{-x \left (t \right )}+{\mathrm e}^{-2 x \left (t \right )} \cos \left (y \left (t \right )\right )^{2}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&={\mathrm e}^{-2 x \left (t \right )} \sin \left (y \left (t \right )\right ) \cos \left (y \left (t \right )\right )-\frac {\sin \left (y \left (t \right )\right )}{\cos \left (y \left (t \right )\right )^{3}} \end{align*}

✗ Maple

ode:=[diff(diff(x(t),t),t) = a*exp(2*x(t))-exp(-x(t))+exp(-2*x(t))*cos(y(t))^2, diff(diff(y(t),t),t) = exp(-2*x(t))*sin(y(t))*cos(y(t))-sin(y(t))/cos(y(t))^3]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],{t,2}]==a*Exp[2*x[t]]-Exp[-x[t]]+Exp[-2*x[t]]*Cos[y[t]]^2,D[y[t],{t,2}]==Exp[-2*x[t]]*Sin[y[t]]*Cos[y[t]]-Sin[y[t]]/Cos[y[t]]^3}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-a*exp(2*x(t)) + Derivative(x(t), (t, 2)) + exp(-x(t)) - exp(-2*x(t))*cos(y(t))**2,0),Eq(sin(y(t))/cos(y(t))**3 + Derivative(y(t), (t, 2)) - exp(-2*x(t))*sin(y(t))*cos(y(t)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError :