60.9.25 problem 1880
Internal
problem
ID
[11804]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1880
Date
solved
:
Friday, March 14, 2025 at 02:58:21 AM
CAS
classification
:
system_of_ODEs
\begin{align*} t^{2} \left (1-\sin \left (t \right )\right ) \left (\frac {d}{d t}x \left (t \right )\right )&=t \left (1-2 \sin \left (t \right )\right ) x \left (t \right )+t^{2} y \left (t \right )\\ t^{2} \left (1-\sin \left (t \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right )&=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x \left (t \right )+t \left (1-t \cos \left (t \right )\right ) y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.165 (sec). Leaf size: 22
ode:=[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)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= t \left (c_{1} t +c_{2} \right ) \\
y \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} t \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 27
ode={t^2*(1-Sin[t])*D[x[t],t]==t*(1-2*Sin[t])*x[t]+t^2*y[t],t^2*(1-Sin[t])*D[y[t],t]==(t*Cos[t]-Sin[t])*x[t]+t*(1-t*Cos[t])*y[t]};
ic={};
DSolve[{ode,ic},{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*}
✓ Sympy. Time used: 2.704 (sec). Leaf size: 201
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(t**2*(1 - sin(t))*Derivative(x(t), t) - t**2*y(t) - t*(1 - 2*sin(t))*x(t),0),Eq(t**2*(1 - sin(t))*Derivative(y(t), t) - t*(-t*cos(t) + 1)*y(t) - (t*cos(t) - sin(t))*x(t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} x_{0}{\left (t \right )} + C_{2} x_{0}{\left (t \right )} \int \frac {t^{2} \left (e^{\int \frac {- 2 t \sin {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}\right ) e^{\int \frac {- t^{2} \cos {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}}{\left (- t^{2} \sin {\left (t \right )} + t^{2}\right ) x_{0}^{2}{\left (t \right )}}\, dt, \ y{\left (t \right )} = C_{1} y_{0}{\left (t \right )} + C_{2} \left (y_{0}{\left (t \right )} \int \frac {t^{2} \left (e^{\int \frac {- 2 t \sin {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}\right ) e^{\int \frac {- t^{2} \cos {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}}{\left (- t^{2} \sin {\left (t \right )} + t^{2}\right ) x_{0}^{2}{\left (t \right )}}\, dt + \frac {\left (e^{\int \frac {- 2 t \sin {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}\right ) e^{\int \frac {- t^{2} \cos {\left (t \right )} + t}{- t^{2} \sin {\left (t \right )} + t^{2}}\, dt}}{x_{0}{\left (t \right )}}\right )\right ]
\]