Internal problem ID [3385]
Internal file name [OUTPUT/2878_Sunday_June_05_2022_08_45_02_AM_30900568/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 128.
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]
\[ \boxed {y^{\prime }-\sqrt {a +b \cos \left (y\right )}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{\sqrt {a +b \cos \left (y \right )}}d y &= \int d x \\ \frac {2 \sqrt {\frac {a +b \cos \left (y\right )}{a +b}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {b}}{\sqrt {a +b}}\right )}{\sqrt {a +b \cos \left (y\right )}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} \frac {2 \sqrt {\frac {a +b \cos \left (y\right )}{a +b}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {b}}{\sqrt {a +b}}\right )}{\sqrt {a +b \cos \left (y\right )}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \frac {2 \sqrt {\frac {a +b \cos \left (y\right )}{a +b}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {b}}{\sqrt {a +b}}\right )}{\sqrt {a +b \cos \left (y\right )}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {a +b \cos \left (y\right )}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {a +b \cos \left (y\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {a +b \cos \left (y\right )}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {a +b \cos \left (y\right )}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 \sqrt {\frac {a +b \cos \left (y\right )}{a +b}}\, \mathrm {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {b}}{\sqrt {a +b}}\right )}{\sqrt {a +b \cos \left (y\right )}}=x +c_{1} \end {array} \]
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: 21
dsolve(diff(y(x),x) = sqrt(a+b*cos(y(x))),y(x), singsol=all)
\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a +b \cos \left (\textit {\_a} \right )}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.83 (sec). Leaf size: 55
DSolve[y'[x]==Sqrt[a+b Cos[ y[x]]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {a+b} (x+c_1),\frac {2 b}{a+b}\right ) \\ y(x)\to -\arccos \left (-\frac {a}{b}\right ) \\ y(x)\to \arccos \left (-\frac {a}{b}\right ) \\ \end{align*}