79.9.1 problem (a)
Internal
problem
ID
[21107]
Book
:
Ordinary
Differential
Equations.
By
Wolfgang
Walter.
Graduate
texts
in
Mathematics.
Springer.
NY.
QA372.W224
1998
Section
:
Chapter
IV.
Linear
Differential
Equations.
Excercises
IV
at
page
172
Problem
number
:
(a)
Date
solved
:
Sunday, October 12, 2025 at 05:51:26 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right ) \cos \left (t \right )-y \left (t \right ) \sin \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \sin \left (t \right )+y \left (t \right ) \cos \left (t \right ) \end{align*}
✓ Maple. Time used: 0.358 (sec). Leaf size: 57
ode:=[diff(x(t),t) = x(t)*cos(t)-y(t)*sin(t), diff(y(t),t) = x(t)*sin(t)+y(t)*cos(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{i \cos \left (t \right )+\sin \left (t \right )}+c_2 \,{\mathrm e}^{-i \cos \left (t \right )+\sin \left (t \right )} \\
y \left (t \right ) &= i \left (c_1 \,{\mathrm e}^{i \cos \left (t \right )+\sin \left (t \right )}-c_2 \,{\mathrm e}^{-i \cos \left (t \right )+\sin \left (t \right )}\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 45
ode={D[x[t],t]==x[t]*Cos[t]-y[t]*Sin[t], D[y[t],t]==x[t]*Sin[t]+y[t]*Cos[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to e^{\sin (t)} (c_1 \cos (\cos (t))+c_2 \sin (\cos (t)))\\ y(t)&\to e^{\sin (t)} (c_2 \cos (\cos (t))-c_1 \sin (\cos (t))) \end{align*}
✓ Sympy. Time used: 0.229 (sec). Leaf size: 207
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t)*cos(t) + y(t)*sin(t) + Derivative(x(t), t),0),Eq(-x(t)*sin(t) - y(t)*cos(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} e^{- \sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}} \cos {\left (t \right )}}{2 \sqrt {\sin ^{2}{\left (t \right )} - 1}} + \frac {C_{1} e^{\sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}} \cos {\left (t \right )}}{2 \sqrt {\sin ^{2}{\left (t \right )} - 1}} + \frac {C_{2} e^{- \sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2} + \frac {C_{2} e^{\sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2}, \ y{\left (t \right )} = \frac {C_{1} e^{- \sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2} + \frac {C_{1} e^{\sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2} - \frac {C_{2} \sqrt {\sin ^{2}{\left (t \right )} - 1} e^{- \sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2 \cos {\left (t \right )}} + \frac {C_{2} \sqrt {\sin ^{2}{\left (t \right )} - 1} e^{\sqrt {\sin ^{2}{\left (t \right )} - 1} + \sin {\left (t \right )}}}{2 \cos {\left (t \right )}}\right ]
\]