29.5.12 problem 128

Internal problem ID [4729]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 5
Problem number : 128
Date solved : Monday, January 27, 2025 at 09:34:38 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\sqrt {a +b \cos \left (y\right )} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 21

dsolve(diff(y(x),x) = sqrt(a+b*cos(y(x))),y(x), singsol=all)
 

\[ x -\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a +b \cos \left (\textit {\_a} \right )}}d \textit {\_a} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.724 (sec). Leaf size: 55

DSolve[D[y[x],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*}