Internal problem ID [10212]
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.
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*}
✓ Solution by Maple
Time used: 8.032 (sec). Leaf size: 648
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)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = c_{2} t \left (\int \left (\frac {i \left ({\mathrm e}^{2 i t}\right )^{2} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}+\frac {\left ({\mathrm e}^{2 i t}\right )^{2} \left (t^{2}+1\right ) t}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}-\frac {i {\mathrm e}^{2 i t} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}+\frac {{\mathrm e}^{2 i t} \left (t^{2}+1\right ) t}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}\right ) \cos \left (t \right )d t \right )+\frac {t \left (\sin \left (t \right )^{2} \cos \left (t \right ) \left (\frac {\left ({\mathrm e}^{2 i t}\right )^{2} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2}}-\frac {i {\mathrm e}^{2 i t} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}+\frac {{\mathrm e}^{2 i t} \left (t^{2}+1\right ) t}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}\right ) c_{2} t -\sin \left (t \right ) \cos \left (t \right ) \left (\frac {\left ({\mathrm e}^{2 i t}\right )^{2} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2}}-\frac {i {\mathrm e}^{2 i t} \left (t^{2}+1\right )}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}+\frac {{\mathrm e}^{2 i t} \left (t^{2}+1\right ) t}{\left (i t +1\right ) \left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}\right ) c_{2} t -\cos \left (t \right ) c_{1} t +c_{1} \sin \left (t \right )\right )}{-t \cos \left (t \right )+\sin \left (t \right )} \] \[ y \left (t \right ) = \sin \left (t \right ) \left (\left (\int \frac {\cos \left (t \right ) \sqrt {t^{2}+1}\, \left (i {\mathrm e}^{-i \arctan \left (t \right )+4 i t}-i {\mathrm e}^{2 i t -i \arctan \left (t \right )}+t \,{\mathrm e}^{-i \arctan \left (t \right )+4 i t}+t \,{\mathrm e}^{2 i t -i \arctan \left (t \right )}\right )}{\left ({\mathrm e}^{2 i t}+1\right ) \left ({\mathrm e}^{2 i t}-1\right )^{2} \left (t +i\right )}d t \right ) c_{2} +c_{1} \right ) \]
✓ 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*}