60.10.5 problem 1917
Internal
problem
ID
[11840]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1917
Date
solved
:
Friday, March 14, 2025 at 03:00:13 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )^{2}-\cos \left (x \left (t \right )\right )\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right ) \sin \left (x \left (t \right )\right ) \end{align*}
✓ Maple. Time used: 2.285 (sec). Leaf size: 105
ode:=[diff(x(t),t) = y(t)^2-cos(x(t)), diff(y(t),t) = -y(t)*sin(x(t))];
dsolve(ode);
\begin{align*}
\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{3 \tan \left (\operatorname {RootOf}\left (-3 \sqrt {-\cos \left (\textit {\_f} \right )^{2}}\, \ln \left (\frac {9 \cos \left (\textit {\_f} \right )^{2} \tan \left (\textit {\_Z} \right )^{2}}{4}+\frac {9 \cos \left (\textit {\_f} \right )^{2}}{4}\right )+c_{1} \sqrt {-\cos \left (\textit {\_f} \right )^{2}}-2 \cos \left (\textit {\_f} \right ) \textit {\_Z} \right )\right ) \sqrt {-\cos \left (\textit {\_f} \right )^{2}}+\cos \left (\textit {\_f} \right )}d \textit {\_f} \right )+t +c_{2} \right )\right \} \\
\left \{y \left (t \right ) &= \sqrt {\frac {d}{d t}x \left (t \right )+\cos \left (x \left (t \right )\right )}, y \left (t \right ) = -\sqrt {\frac {d}{d t}x \left (t \right )+\cos \left (x \left (t \right )\right )}\right \} \\
\end{align*}
✓ Mathematica. Time used: 0.373 (sec). Leaf size: 399
ode={D[x[t],t]==y[t]^2-Cos[x[t]],D[y[t],t]==-y[t]*Sin[x[t]]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -\arccos \left (\frac {\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \sqrt {\frac {-K[2]^6+6 c_1 K[2]^3+9 K[2]^2-9 c_1{}^2}{K[2]^2}}}dK[2]\&\right ]\left [\frac {t}{3}+c_2\right ]{}^3-3 c_1}{3 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \sqrt {\frac {-K[2]^6+6 c_1 K[2]^3+9 K[2]^2-9 c_1{}^2}{K[2]^2}}}dK[2]\&\right ]\left [\frac {t}{3}+c_2\right ]}\right ) \\
y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \sqrt {\frac {-K[2]^6+6 c_1 K[2]^3+9 K[2]^2-9 c_1{}^2}{K[2]^2}}}dK[2]\&\right ]\left [\frac {t}{3}+c_2\right ] \\
x(t)\to \arccos \left (\frac {\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3] \sqrt {\frac {-K[3]^6+6 c_1 K[3]^3+9 K[3]^2-9 c_1{}^2}{K[3]^2}}}dK[3]\&\right ]\left [-\frac {t}{3}+c_2\right ]{}^3-3 c_1}{3 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3] \sqrt {\frac {-K[3]^6+6 c_1 K[3]^3+9 K[3]^2-9 c_1{}^2}{K[3]^2}}}dK[3]\&\right ]\left [-\frac {t}{3}+c_2\right ]}\right ) \\
y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3] \sqrt {\frac {-K[3]^6+6 c_1 K[3]^3+9 K[3]^2-9 c_1{}^2}{K[3]^2}}}dK[3]\&\right ]\left [-\frac {t}{3}+c_2\right ] \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-y(t)**2 + cos(x(t)) + Derivative(x(t), t),0),Eq(y(t)*sin(x(t)) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
ZeroDivisionError : polynomial division