Internal problem ID [10519]
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]
\[ \boxed {y^{\prime }-\left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}=-a +\lambda +a \cos \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 102
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)
\[ y \left (x \right ) = \frac {2 \sec \left (x \lambda \right )^{2} {\mathrm e}^{-\frac {a \cos \left (2 x \lambda \right )}{2 \lambda }} c_{1} \lambda -2 \tan \left (x \lambda \right ) \lambda \left (\int {\mathrm e}^{-\frac {a \cos \left (2 x \lambda \right )}{2 \lambda }} \left (\sec \left (x \lambda \right )^{2} \lambda +a \right )d x \right ) c_{1} +i \tan \left (x \lambda \right )}{-2 \lambda \left (\int {\mathrm e}^{-\frac {a \cos \left (2 x \lambda \right )}{2 \lambda }} \left (\sec \left (x \lambda \right )^{2} \lambda +a \right )d x \right ) c_{1} +i} \]
✓ 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*}