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
:
Monday, January 27, 2025 at 09:29:42 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 96
dsolve(diff(y(x),x) = cos(2*x)+(sin(2*x)+y(x))*y(x),y(x), singsol=all)
\[
y \left (x \right ) = \frac {\sin \left (x \right ) \left (\operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \cos \left (x \right ) \left (\cos \left (x \right ) \operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} +\operatorname {HeunCPrime}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )\right )}{c_{1} \cos \left (x \right ) \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}
\]
✓ Solution by Mathematica
Time used: 2.195 (sec). Leaf size: 111
DSolve[D[y[x],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*}