54.10.5 problem 1917
Internal
problem
ID
[13139]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1917
Date
solved
:
Sunday, October 12, 2025 at 02:35:47 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: 0.775 (sec). Leaf size: 107
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 (-\int _{}^{\textit {\_Z}}\frac {1}{\operatorname {RootOf}\left (21 \ln \left (\frac {126 \cos \left (\textit {\_f} \right )}{\textit {\_Z}}\right )-7 \ln \left (\frac {\frac {126 \textit {\_Z}}{5}+\frac {126 \cos \left (\textit {\_f} \right )}{5}}{\textit {\_Z}}\right )-14 \ln \left (\frac {-\frac {63 \textit {\_Z}}{4}+\frac {63 \cos \left (\textit {\_f} \right )}{2}}{\textit {\_Z}}\right )-21 \ln \left ({\mathrm e}^{2 i \textit {\_f}}+1\right )+21 \ln \left ({\mathrm e}^{i \textit {\_f}}\right )+18 c_1 \right )}d \textit {\_f} +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: 0.249 (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)