Internal problem ID [3313]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 2
Problem number: 49.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\left (\sin \left (2 x \right )+y\right ) y=\cos \left (2 x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 198
dsolve(diff(y(x),x) = cos(2*x)+(sin(2*x)+y(x))*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {2 \operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \cos \left (2 x \right )}{\sqrt {2 \cos \left (2 x \right )+2}\, \left (c_{1} \sqrt {2 \cos \left (2 x \right )+2}\, \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )}+\frac {\operatorname {HeunCPrime}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}+2 \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}}{\sqrt {2 \cos \left (2 x \right )+2}\, \left (c_{1} \sqrt {2 \cos \left (2 x \right )+2}\, \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )}\right ) \sin \left (2 x \right ) \]
✓ Solution by Mathematica
Time used: 2.305 (sec). Leaf size: 111
DSolve[y'[x]==Cos[2 x]+(Sin[2 x]+y[x])y[x],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*}