54.10.9 problem 1923

Internal problem ID [13143]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1923
Date solved : Sunday, October 12, 2025 at 02:35:47 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.107 (sec). Leaf size: 34

ode:=[(t^2+1)*diff(x(t),t) = -t*x(t)+y(t), (t^2+1)*diff(y(t),t) = -x(t)-t*y(t)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 107

ode={(t^2+1)*D[x[t],t]==-t*x[t]+y[t],(t^2+1)*D[y[t],t]==-x[t]-t*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 99

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(t*x(t) + (t**2 + 1)*Derivative(x(t), t) - y(t),0),Eq(t*y(t) + (t**2 + 1)*Derivative(y(t), t) + x(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {\left (\frac {C_{1}}{2} - \frac {i C_{2}}{2}\right ) e^{i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}} + \frac {\left (\frac {C_{1}}{2} + \frac {i C_{2}}{2}\right ) e^{- i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}}, \ y{\left (t \right )} = - \frac {\left (\frac {i C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}} + \frac {\left (\frac {i C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{i \operatorname {atan}{\left (t \right )}}}{\sqrt {t^{2} + 1}}\right ] \]