Internal problem ID [7603]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 198
dsolve(diff(y(x),x) - y(x)^2 -y(x)*sin(2*x) - cos(2*x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\frac {2 \HeunCPrime \left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \cos \left (2 x \right )}{\sqrt {2 \cos \left (2 x \right )+2}\, \left (c_{1} \HeunC \left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}+\HeunC \left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )}+\frac {\HeunCPrime \left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}+2 \HeunCPrime \left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}+2 \HeunC \left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}}{\sqrt {2 \cos \left (2 x \right )+2}\, \left (c_{1} \HeunC \left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}+\HeunC \left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )}\right ) \sin \left (2 x \right ) \]
✓ Solution by Mathematica
Time used: 1.631 (sec). Leaf size: 73
DSolve[y'[x] - y[x]^2 -y[x]*Sin[2*x] - Cos[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \tan (x)+\frac {e^{-\cos ^2(x)} \tan (x) \sec (x)}{\sqrt {-\sin ^2(x)} \left (\int _1^{\cos (x)}\frac {e^{-K[1]^2}}{K[1]^2 \sqrt {K[1]^2-1}}dK[1]+c_1\right )} \\ y(x)\to \tan (x) \\ \end{align*}