60.1.511 problem 514
Internal
problem
ID
[10525]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
514
Date
solved
:
Monday, January 27, 2025 at 08:51:33 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*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 84
dsolve(diff(y(x),x)^2*(a*cos(y(x))+b)-c*cos(y(x))+d=0,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*}
✓ Solution by Mathematica
Time used: 11.716 (sec). Leaf size: 627
DSolve[d - c*Cos[y[x]] + (b + a*Cos[y[x]])*D[y[x],x]^2==0,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*}