61.13.11 problem 57
Internal
problem
ID
[12231]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-5.
Equations
containing
combinations
of
trigonometric
functions.
Problem
number
:
57
Date
solved
:
Tuesday, January 28, 2025 at 01:30:40 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 268
dsolve(diff(y(x),x)=y(x)^2+lambda*a+lambda*b+2*a*b+a*(lambda-a)*tan(lambda*x)^2+b*(lambda-b)*cot(lambda*x)^2,y(x), singsol=all)
\[
y = \frac {4 c_{1} \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (b -\lambda +a \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -b -a}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 c_{1} \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (\lambda x \right )^{2}+a^{2} \sin \left (\lambda x \right )^{2}-\frac {5 \left (a -\frac {3 \lambda }{5}\right ) \lambda }{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \left (a -\frac {3 \lambda }{2}\right ) \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (\lambda x \right )-\cot \left (\lambda x \right ) b \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_{1} \cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}\right )}
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[D[y[x],x]==y[x]^2+\[Lambda]*a+\[Lambda]*b+2*a*b+a*(\[Lambda]-a)*Tan[\[Lambda]*x]^2+b*(\[Lambda]-b)*Cot[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved