61.13.4 problem 50
Internal
problem
ID
[12224]
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
:
50
Date
solved
:
Tuesday, January 28, 2025 at 01:25:39 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \cos \left (\lambda x \right ) y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 1066
dsolve(diff(y(x),x)=a*cos(lambda*x)*y(x)^2+b*cos(lambda*x)*sin(lambda*x)^n,y(x), singsol=all)
\[
\text {Expression too large to display}
\]
✓ Solution by Mathematica
Time used: 0.656 (sec). Leaf size: 633
DSolve[D[y[x],x]==a*Cos[\[Lambda]*x]*y[x]^2+b*Cos[\[Lambda]*x]*Sin[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\csc (\lambda x) \left (-\lambda \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(\lambda x) \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \left (\operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )-c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{2 a \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \sin ^{\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} \sin ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}-\lambda \csc (\lambda x)}{2 a} \\
\end{align*}