Internal problem ID [8696]
Internal file name [OUTPUT/7629_Sunday_June_05_2022_11_23_09_PM_32218282/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 360.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos \left (a y\right )^{2}-1+c \cos \left (a y\right )}=0} \]
Integrating both sides gives \begin {align*} \int -\frac {\cos \left (a y \right )}{b \sqrt {\cos \left (a y \right )^{2}-1+c \cos \left (a y \right )}\, \left (c \cos \left (a y \right )-1\right )}d y &= \int {dx}\\ \int _{}^{y}-\frac {\cos \left (a \textit {\_a} \right )}{b \sqrt {\cos \left (a \textit {\_a} \right )^{2}-1+c \cos \left (a \textit {\_a} \right )}\, \left (c \cos \left (a \textit {\_a} \right )-1\right )}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {\cos \left (a \textit {\_a} \right )}{b \sqrt {\cos \left (a \textit {\_a} \right )^{2}-1+c \cos \left (a \textit {\_a} \right )}\, \left (c \cos \left (a \textit {\_a} \right )-1\right )}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {\cos \left (a \textit {\_a} \right )}{b \sqrt {\cos \left (a \textit {\_a} \right )^{2}-1+c \cos \left (a \textit {\_a} \right )}\, \left (c \cos \left (a \textit {\_a} \right )-1\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(diff(y(x),x)*cos(a*y(x))-b*(1-c*cos(a*y(x)))*(cos(a*y(x))^2-1+c*cos(a*y(x)))^(1/2) = 0,y(x), singsol=all)
\[ \frac {\int _{}^{y \left (x \right )}\frac {\cos \left (\textit {\_a} a \right )}{\sqrt {c \cos \left (\textit {\_a} a \right )-\sin \left (\textit {\_a} a \right )^{2}}\, \left (c \cos \left (\textit {\_a} a \right )-1\right )}d \textit {\_a} +\left (x +c_{1} \right ) b}{b} = 0 \]
✓ Solution by Mathematica
Time used: 28.047 (sec). Leaf size: 504
DSolve[-(b*(1 - c*Cos[a*y[x]])*Sqrt[-1 + c*Cos[a*y[x]] + Cos[a*y[x]]^2]) + Cos[a*y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {i (\cos (\text {$\#$1} a)+1) \sqrt {\frac {2 c \cos (\text {$\#$1} a)+\cos (2 \text {$\#$1} a)-1}{(\cos (\text {$\#$1} a)+1)^2}} \sqrt {\frac {c \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )+\sqrt {c^2+4}+2}{\sqrt {c^2+4}+2}} \sqrt {1-\frac {c \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )}{\sqrt {c^2+4}-2}} \left ((c-1) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \tan \left (\frac {a \text {$\#$1}}{2}\right )\right ),\frac {2-\sqrt {c^2+4}}{\sqrt {c^2+4}+2}\right )+2 \operatorname {EllipticPi}\left (\frac {(c+1) \left (\sqrt {c^2+4}-2\right )}{(c-1) c},i \text {arcsinh}\left (\sqrt {-\frac {c}{\sqrt {c^2+4}-2}} \tan \left (\frac {a \text {$\#$1}}{2}\right )\right ),\frac {2-\sqrt {c^2+4}}{\sqrt {c^2+4}+2}\right )\right )}{a \left (c^2-1\right ) \sqrt {\frac {c}{4-2 \sqrt {c^2+4}}} \sqrt {2 c \cos (\text {$\#$1} a)+\cos (2 \text {$\#$1} a)-1} \sqrt {-c \tan ^4\left (\frac {\text {$\#$1} a}{2}\right )-4 \tan ^2\left (\frac {\text {$\#$1} a}{2}\right )+c}}\&\right ]\left [-\frac {b x}{\sqrt {2}}+c_1\right ] \\ y(x)\to -\frac {\arccos \left (\frac {1}{c}\right )}{a} \\ y(x)\to \frac {\arccos \left (\frac {1}{c}\right )}{a} \\ y(x)\to -\frac {\arccos \left (\frac {1}{2} \left (-\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to \frac {\arccos \left (\frac {1}{2} \left (-\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to -\frac {\arccos \left (\frac {1}{2} \left (\sqrt {c^2+4}-c\right )\right )}{a} \\ y(x)\to \frac {\arccos \left (\frac {1}{2} \left (\sqrt {c^2+4}-c\right )\right )}{a} \\ \end{align*}