23.1.110 problem 111

Internal problem ID [4717]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 111
Date solved : Tuesday, September 30, 2025 at 08:19:21 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: 0.524 (sec). Leaf size: 377

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {a \int _1^x\left (\frac {b B^3 (a+b \cos (A K[1]+B K[2])) \sin (A K[1]+B K[2])}{(A+a B+b B \cos (A K[1]+B K[2]))^2}-\frac {b B^2 \sin (A K[1]+B K[2])}{A+a B+b B \cos (A K[1]+B K[2])}\right )dK[1] B+b \cos (A x+B K[2]) \int _1^x\left (\frac {b B^3 (a+b \cos (A K[1]+B K[2])) \sin (A K[1]+B K[2])}{(A+a B+b B \cos (A K[1]+B K[2]))^2}-\frac {b B^2 \sin (A K[1]+B K[2])}{A+a B+b B \cos (A K[1]+B K[2])}\right )dK[1] B+B+A \int _1^x\left (\frac {b B^3 (a+b \cos (A K[1]+B K[2])) \sin (A K[1]+B K[2])}{(A+a B+b B \cos (A K[1]+B K[2]))^2}-\frac {b B^2 \sin (A K[1]+B K[2])}{A+a B+b B \cos (A K[1]+B K[2])}\right )dK[1]}{A+a B+b B \cos (A x+B K[2])}dK[2]+\int _1^x\frac {B (a+b \cos (A K[1]+B y(x)))}{A+a B+b B \cos (A K[1]+B y(x))}dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 117.681 (sec). Leaf size: 1600

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)
 

\[ \text {Solution too large to show} \]