Internal problem ID [10529]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with
cosine.
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\left (\lambda +\cos \left (\lambda x \right )^{2} a \right ) y^{2}=-a +\lambda +\cos \left (\lambda x \right )^{2} a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 550
dsolve(diff(y(x),x)=(lambda+a*cos(lambda*x)^2)*y(x)^2+lambda-a+a*cos(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {\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 }}}{2 \left (1+\cos \left (2 \lambda x \right )\right )^{2} \sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) \left (\left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right ) \lambda }{\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right )}+\frac {\left (\left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right ) \lambda }{\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a +a \sqrt {-1+\cos \left (2 \lambda x \right )}\right ) \cos \left (2 \lambda x \right )^{2}+\left (\left (2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a +2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right ) \lambda }{\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right )+2 a \sqrt {-1+\cos \left (2 \lambda x \right )}+2 \lambda \sqrt {-1+\cos \left (2 \lambda x \right )}\right ) \cos \left (2 \lambda x \right )+\left (\sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a +2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right ) \lambda }{\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right )+a \sqrt {-1+\cos \left (2 \lambda x \right )}+2 \lambda \sqrt {-1+\cos \left (2 \lambda x \right )}}{2 \left (1+\cos \left (2 \lambda x \right )\right )^{2} \sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) \left (\left (\int -\frac {2 \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \sin \left (2 \lambda x \right ) \lambda }{\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (1+\cos \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right )}\right ) \sin \left (2 \lambda x \right ) \]
✓ Solution by Mathematica
Time used: 36.333 (sec). Leaf size: 263
DSolve[y'[x]==(\[Lambda]+a*Cos[\[Lambda]*x]^2)*y[x]^2+\[Lambda]-a+a*Cos[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \left (c_1 \tan (\lambda x) \int _1^xe^{-\frac {a \cos ^2(\lambda K[1])}{\lambda }} \left (\lambda \sec ^2(\lambda K[1])+a\right )dK[1]+c_1 \sec ^2(\lambda x) \left (-e^{-\frac {a \cos ^2(\lambda x)}{\lambda }}\right )+\tan (\lambda x)\right )}{2+2 c_1 \int _1^xe^{-\frac {a \cos ^2(\lambda K[1])}{\lambda }} \left (\lambda \sec ^2(\lambda K[1])+a\right )dK[1]} y(x)\to \frac {1}{2} \sec ^2(\lambda x) \left (\sin (2 \lambda x)-\frac {2 e^{-\frac {a \cos ^2(\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {a \cos ^2(\lambda K[1])}{\lambda }} \left (\lambda \sec ^2(\lambda K[1])+a\right )dK[1]}\right ) y(x)\to \frac {1}{2} \sec ^2(\lambda x) \left (\sin (2 \lambda x)-\frac {2 e^{-\frac {a \cos ^2(\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {a \cos ^2(\lambda K[1])}{\lambda }} \left (\lambda \sec ^2(\lambda K[1])+a\right )dK[1]}\right ) \end{align*}