Internal problem ID [9763]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with sine
Problem number: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\left (\lambda +\left (\sin ^{2}\left (\lambda x \right )\right ) a \right ) y^{2}-\lambda +a -\left (\sin ^{2}\left (\lambda x \right )\right ) a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 467
dsolve(diff(y(x),x)=(lambda+a*sin(lambda*x)^2)*y(x)^2+lambda-a+a*sin(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (-4 \cos \left (2 \lambda x \right ) \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a \lambda +4 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a \lambda +8 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} \lambda ^{2}\right ) {\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }}+\left (\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (\int -\frac {2 \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1} a +a \sqrt {\cos \left (2 \lambda x \right )+1}\right ) \left (\cos ^{2}\left (2 \lambda x \right )\right )+\left (\left (-2 \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} a -2 \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} \lambda \right ) \left (\int -\frac {2 \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right )-2 a \sqrt {\cos \left (2 \lambda x \right )+1}-2 \lambda \sqrt {\cos \left (2 \lambda x \right )+1}\right ) \cos \left (2 \lambda x \right )+\left (\sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} a +2 \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} \lambda \right ) \left (\int -\frac {2 \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right )+a \sqrt {\cos \left (2 \lambda x \right )+1}+2 \lambda \sqrt {\cos \left (2 \lambda x \right )+1}\right ) \sin \left (2 \lambda x \right )}{2 \left (-1+\cos \left (2 \lambda x \right )\right )^{2} \sqrt {\cos \left (2 \lambda x \right )+1}\, \left (\lambda +\left (\sin ^{2}\left (\lambda x \right )\right ) a \right ) \left (\left (\int -\frac {2 \,{\mathrm e}^{\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )-a -2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\sqrt {\cos \left (2 \lambda x \right )+1}\, \left (-1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1}+1\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==(\[Lambda]+a*Sin[\[Lambda]*x]^2)*y[x]^2+\[Lambda]-a+a*Sin[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved