60.9.21 problem 1876

Internal problem ID [11800]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1876
Date solved : Friday, March 14, 2025 at 02:58:19 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.336 (sec). Leaf size: 17

ode:=[diff(x(t),t) = x(t)*cos(t), diff(y(t),t) = x(t)*exp(-sin(t))]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{\sin \left (t \right )} \\ y \left (t \right ) &= c_{2} t +c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 55

ode={D[x[t],t]==x[t]*Cos[t],D[y[t],t]==x[t]*Exp[-Sin[t]]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 \exp \left (\int _1^t\cos (K[1])dK[1]\right ) \\ y(t)\to c_1 \int _1^t\exp \left (\int _1^{K[2]}\cos (K[1])dK[1]-\sin (K[2])\right )dK[2]+c_2 \\ \end{align*}

✓ Sympy. Time used: 1.181 (sec). Leaf size: 15

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

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