Internal problem ID [9803]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2} \sin \left (\lambda x \right ) a -b \sin \left (\lambda x \right ) \left (\cos ^{n}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 334
dsolve(diff(y(x),x)=a*sin(lambda*x)*y(x)^2+b*sin(lambda*x)*cos(lambda*x)^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {-\frac {\sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right ) c_{1} \lambda \BesselY \left (\frac {n +3}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right )}{\left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right )\right ) a}-\frac {\left (\BesselJ \left (\frac {n +3}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )-\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right ) c_{1}-\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right )\right ) \lambda }{\left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \left (\cos ^{1+\frac {n}{2}}\left (\lambda x \right )\right )}{2+n}\right )\right ) a}}{\cos \left (\lambda x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==a*Sin[\[Lambda]*x]*y[x]^2+b*Sin[\[Lambda]*x]*Cos[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved