60.9.19 problem 1874

Internal problem ID [11873]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1874
Date solved : Tuesday, January 28, 2025 at 06:23:55 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right ) f \left (t \right )+y \left (t \right ) g \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right ) g \left (t \right )+y \left (t \right ) f \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.871 (sec). Leaf size: 56

dsolve([diff(x(t),t)=x(t)*f(t)+y(t)*g(t),diff(y(t),t)=-x(t)*g(t)+y(t)*f(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\int \left (\tan \left (c_{1} -\int g \left (t \right )d t \right ) g \left (t \right )+f \left (t \right )\right )d t} c_{2} \\ y \left (t \right ) &= {\mathrm e}^{\int \left (\tan \left (c_{1} -\int g \left (t \right )d t \right ) g \left (t \right )+f \left (t \right )\right )d t} c_{2} \tan \left (c_{1} -\int g \left (t \right )d t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 93

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

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