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 : Sunday, March 30, 2025 at 03:46:50 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 74

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

\[ y = \frac {-A x -2 \arctan \left (\frac {\tan \left (\frac {\sqrt {\left (A +\left (a +b \right ) B \right ) \left (A +\left (a -b \right ) B \right )}\, \left (c_1 -x \right )}{2}\right ) \sqrt {\left (A +\left (a +b \right ) B \right ) \left (A +\left (a -b \right ) B \right )}}{A +\left (a -b \right ) B}\right )}{B} \]

✓ Mathematica. Time used: 60.71 (sec). Leaf size: 102

ode=D[y[x],x]==a+b Cos[A x+ B y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a - b*cos(A*x + B*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : no valid subset found