Internal problem ID [10202]
Internal file name [OUTPUT/9149_Monday_June_06_2022_06_45_16_AM_13118382/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1880.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {2 x \left (t \right ) \sin \left (t \right )}{t \left (\sin \left (t \right )-1\right )}-\frac {y \left (t \right )}{\sin \left (t \right )-1}-\frac {x \left (t \right )}{t \left (\sin \left (t \right )-1\right )}\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right ) \cos \left (t \right )}{\sin \left (t \right )-1}-\frac {x \left (t \right ) \cos \left (t \right )}{t \left (\sin \left (t \right )-1\right )}+\frac {x \left (t \right ) \sin \left (t \right )}{t^{2} \left (\sin \left (t \right )-1\right )}-\frac {y \left (t \right )}{t \left (\sin \left (t \right )-1\right )} \end {align*}
Does not currently support non autonomous system of first order linear differential equations. The following is the phase plot
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 23
dsolve([t^2*(1-sin(t))*diff(x(t),t)=t*(1-2*sin(t))*x(t)+t^2*y(t),t^2*(1-sin(t))*diff(y(t),t)=(t*cos(t)-sin(t))*x(t)+t*(1-t*cos(t))*y(t)],singsol=all)
\begin{align*} x \left (t \right ) &= t \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} t \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 27
DSolve[{t^2*(1-Sin[t])*x'[t]==t*(1-2*Sin[t])*x[t]+t^2*y[t],t^2*(1-Sin[t])*y'[t]==(t*Cos[t]-Sin[t])*x[t]+t*(1-t*Cos[t])*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to t (c_1 t+c_2) \\ y(t)\to c_1 t+c_2 \sin (t) \\ \end{align*}