54.1.22 problem 22

Internal problem ID [11336]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 07:49:20 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 96

ode:=diff(y(x),x)-y(x)^2-y(x)*sin(2*x)-cos(2*x) = 0; 
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: 1.339 (sec). Leaf size: 238

ode=D[y[x],x] - y[x]^2 -y[x]*Sin[2*x] - Cos[2*x]==0; 
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)**2 - y(x)*sin(2*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 lie group method