Internal problem ID [9464]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1885.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} t x^{\prime \prime }\relax (t )+2 x^{\prime }\relax (t )+x \relax (t ) t&=0\\ t x^{\prime }\relax (t )-y^{\prime }\relax (t ) t -2 y \relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.1 (sec). Leaf size: 47
dsolve({t*diff(x(t),t)-t*diff(y(t),t)-2*y(t)=0,t*diff(x(t),t,t)+2*diff(x(t),t)+t*x(t)=0},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = \frac {\sin \relax (t ) c_{2}+c_{3} \cos \relax (t )}{t} \] \[ y \relax (t ) = \frac {\sin \relax (t ) c_{2} t +\cos \relax (t ) c_{3} t -2 c_{3} \sin \relax (t )+2 c_{2} \cos \relax (t )+c_{1}}{t^{2}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 54
DSolve[{t*x'[t]-t*y'[t]-2*y[t]==0,t*x''[t]+2*x'[t]+t*x[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {c_2 \cos (t)+c_3 \sin (t)}{t} \\ y(t)\to \frac {c_2 t \cos (t)+2 c_3 \cos (t)-2 c_2 \sin (t)+c_3 t \sin (t)+c_1}{t^2} \\ \end{align*}