61.13.2 problem 48
Internal
problem
ID
[12222]
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
:
48
Date
solved
:
Tuesday, January 28, 2025 at 01:24:32 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 256
dsolve(diff(y(x),x)=a*sin(lambda*x)*y(x)^2+b*sin(lambda*x)*cos(lambda*x)^n,y(x), singsol=all)
\[
y = \frac {\left (-\sqrt {\frac {a b}{\lambda ^{2}}}\, \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) \cos \left (\lambda x \right )^{\frac {n}{2}+1} c_{1} -\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) \sec \left (\lambda x \right ) \lambda }{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a}
\]
✓ Solution by Mathematica
Time used: 0.728 (sec). Leaf size: 695
DSolve[D[y[x],x]==a*Sin[\[Lambda]*x]*y[x]^2+b*Sin[\[Lambda]*x]*Cos[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\lambda \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}{2 a \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+2 a c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}+\lambda \sec (\lambda x)}{2 a} \\
\end{align*}