29.4.20 problem 109

Internal problem ID [4711]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 109
Date solved : Monday, January 27, 2025 at 09:32:55 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 74

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

\[ y \left (x \right ) = \frac {-A x -2 \arctan \left (\frac {\tan \left (\frac {\sqrt {\left (A +\left (a +b \right ) B \right ) \left (A +\left (a -b \right ) B \right )}\, \left (-x +c_{1} \right )}{2}\right ) \sqrt {\left (A +\left (a +b \right ) B \right ) \left (A +\left (a -b \right ) B \right )}}{A +\left (a -b \right ) B}\right )}{B} \]

✓ Solution by Mathematica

Time used: 60.710 (sec). Leaf size: 102

DSolve[D[y[x],x]==a+b Cos[A x+ B y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {A x+2 \arctan \left (\frac {(B (a+b)+A) \tanh \left (\frac {(x-c_1) \left (B^2 \left (a^2-b^2\right )+2 a A B+A^2\right )}{2 \sqrt {-((B (a-b)+A) (B (a+b)+A))}}\right )}{\sqrt {-((B (a-b)+A) (B (a+b)+A))}}\right )}{B} \]