60.9.21 problem 1876

Internal problem ID [11875]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1876
Date solved : Tuesday, January 28, 2025 at 06:23:56 PM
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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t)=x(t)*cos(t),diff(y(t),t)=x(t)*exp(-sin(t))],singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==x[t]*Cos[t],D[y[t],t]==x[t]*Exp[-Sin[t]]},{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*}