54.1.501 problem 514
Internal
problem
ID
[11815]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
514
Date
solved
:
Tuesday, September 30, 2025 at 11:01:45 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d&=0 \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 84
ode:=diff(y(x),x)^2*(a*cos(y(x))+b)-c*cos(y(x))+d = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \arccos \left (\frac {d}{c}\right ) \\
x -\int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_1 &= 0 \\
x +\int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 7.979 (sec). Leaf size: 627
ode=d - c*Cos[y[x]] + (b + a*Cos[y[x]])*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \operatorname {EllipticPi}\left (\frac {b c+a d}{a c+b c},\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][-x+c_1]\\ y(x)&\to \text {InverseFunction}\left [\frac {4 \sin ^2\left (\frac {\text {$\#$1}}{2}\right ) \csc (\text {$\#$1}) \sqrt {a \cos (\text {$\#$1})+b} \sqrt {\frac {\cot ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d)}{c+d}} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (a+b) (d-c \cos (\text {$\#$1}))}{a d+b c}} \left (c (a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )+a (d-c) \operatorname {EllipticPi}\left (\frac {b c+a d}{a c+b c},\arcsin \left (\frac {\sqrt {\frac {(a+b) (d-c \cos (\text {$\#$1})) \csc ^2\left (\frac {\text {$\#$1}}{2}\right )}{b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c+a d)}{(a+b) (c+d)}\right )\right )}{c (a+b) \sqrt {c \cos (\text {$\#$1})-d} \sqrt {\frac {\csc ^2\left (\frac {\text {$\#$1}}{2}\right ) (c-d) (a \cos (\text {$\#$1})+b)}{a d+b c}}}\&\right ][x+c_1]\\ y(x)&\to -\arccos \left (\frac {d}{c}\right )\\ y(x)&\to \arccos \left (\frac {d}{c}\right ) \end{align*}
✓ Sympy. Time used: 58.025 (sec). Leaf size: 48
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
y = Function("y")
ode = Eq(-c*cos(y(x)) + d + (a*cos(y(x)) + b)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {c \cos {\left (y \right )} - d}{a \cos {\left (y \right )} + b}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {c \cos {\left (y \right )} - d}{a \cos {\left (y \right )} + b}}}\, dy = C_{1} + x\right ]
\]