23.1.56 problem 50

Internal problem ID [4663]
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 : 50
Date solved : Tuesday, September 30, 2025 at 07:44:35 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 96

ode:=diff(y(x),x) = cos(2*x)+(sin(2*x)+y(x))*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sin \left (x \right ) \left (\operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_1 +2 \cos \left (x \right ) \left (\operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right ) c_1 +\operatorname {HeunCPrime}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )\right )}{c_1 \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right )+\operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )} \]

✓ Mathematica. Time used: 1.315 (sec). Leaf size: 238

ode=D[y[x],x]==Cos[2 x]+(Sin[2 x]+y[x])y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\tan (x) \left (\int _1^{\cos (x)}\frac {\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2 K[1]^3}{2-2 K[1]^2}dK[1]\right )}{K[3]^2}dK[3]+\sec (x) \exp \left (-2 \int _1^{\cos (x)}\frac {K[1]-2 K[1]^3}{2-2 K[1]^2}dK[1]\right )+c_1\right )}{\int _1^{\cos (x)}\frac {\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2 K[1]^3}{2-2 K[1]^2}dK[1]\right )}{K[3]^2}dK[3]+c_1}\\ y(x)&\to \tan (x)\\ y(x)&\to \frac {\tan (x) \sec (x) \exp \left (-2 \int _1^{\cos (x)}\frac {K[1]-2 K[1]^3}{2-2 K[1]^2}dK[1]\right )}{\int _1^{\cos (x)}\frac {\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2 K[1]^3}{2-2 K[1]^2}dK[1]\right )}{K[3]^2}dK[3]}+\tan (x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x) - sin(2*x))*y(x) - cos(2*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -y(x)**2 - y(x)*sin(2*x) - cos(2*x) + Derivative(y(x), x) cannot be solved by the factorable group method