60.9.30 problem 1885

Internal problem ID [11884]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1885
Date solved : Tuesday, January 28, 2025 at 06:23:59 PM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )-t \left (\frac {d}{d t}y \left (t \right )\right )-2 y \left (t \right )&=0\\ t \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+2 \frac {d}{d t}x \left (t \right )+x \left (t \right ) t&=0 \end{align*}

✓ Solution by Maple

Time used: 0.134 (sec). Leaf size: 46

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],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {c_{2} \sin \left (t \right )+c_3 \cos \left (t \right )}{t} \\ y \left (t \right ) &= \frac {\sin \left (t \right ) c_{2} t +\cos \left (t \right ) c_3 t -2 c_3 \sin \left (t \right )+2 c_{2} \cos \left (t \right )+c_{1}}{t^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 54

DSolve[{t*D[x[t],t]-t*D[y[t],t]-2*y[t]==0,t*D[x[t],{t,2}]+2*D[x[t],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*}