23.1.124 problem 127

Internal problem ID [4731]
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 : 127
Date solved : Tuesday, September 30, 2025 at 08:20:26 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 85

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

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

✓ Mathematica. Time used: 60.487 (sec). Leaf size: 94

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

\begin{align*} y(x)&\to \frac {-A x+2 \arctan \left (\frac {-b B+\sqrt {B^2 \left (a^2-b^2\right )+2 a A B+A^2} \tan \left (\frac {1}{2} (x-c_1) \sqrt {B^2 \left (a^2-b^2\right )+2 a A B+A^2}\right )}{a B+A}\right )}{B} \end{align*}

✓ Sympy. Time used: 121.245 (sec). Leaf size: 1051

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*sin(A*x + B*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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