29.2.24 problem 49

Internal problem ID [4657]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 2
Problem number : 49
Date solved : Sunday, March 30, 2025 at 03:32:48 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: 2.195 (sec). Leaf size: 111

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 {\sec (x) \left (\sin (x) \int _1^{\cos (x)}\frac {e^{-K[1]^2}}{K[1]^2 \sqrt {K[1]^2-1}}dK[1]+c_1 \sin (x)+\frac {e^{-\cos ^2(x)} \tan (x)}{\sqrt {-\sin ^2(x)}}\right )}{\int _1^{\cos (x)}\frac {e^{-K[1]^2}}{K[1]^2 \sqrt {K[1]^2-1}}dK[1]+c_1} \\ y(x)\to \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