60.9.7 problem 1862

Internal problem ID [11861]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1862
Date solved : Monday, January 27, 2025 at 11:43:55 PM
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 )&=2 x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 41

dsolve({diff(x(t),t)=-y(t),diff(y(t),t)=2*x(t)+2*y(t)},singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 46

DSolve[{D[x[t],t]==-y[t],D[y[t],t]==2*x[t]+2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^t (c_1 \cos (t)-(c_1+c_2) \sin (t)) \\ y(t)\to e^t (2 c_1 \sin (t)+c_2 (\sin (t)+\cos (t))) \\ \end{align*}