60.10.11 problem 1923

Internal problem ID [11922]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1923
Date solved : Tuesday, January 28, 2025 at 06:24:07 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 34

dsolve([(t^2+1)*diff(x(t),t)=-t*x(t)+y(t),(t^2+1)*diff(y(t),t)=-x(t)-t*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 107

DSolve[{(t^2+1)*D[x[t],t]==-t*x[t]+y[t],(t^2+1)*D[y[t],t]==-x[t]-t*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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