Internal problem ID [9777]
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: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (\cos \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 539
dsolve((a*cos(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*cos(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\lambda \left (\left (\left (a^{4}-b \,a^{3}-b^{2} a^{2}+a \,b^{3}\right ) \sin \left (\frac {\lambda x}{2}\right )^{2}+\left (-4 \sqrt {a^{2}-b^{2}}\, a^{2} b +4 \sqrt {a^{2}-b^{2}}\, a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+\left (2 \sqrt {a^{2}-b^{2}}\, c_{1} a^{2}-2 \sqrt {a^{2}-b^{2}}\, c_{1} a b \right ) \sin \left (\lambda x \right )+\left (-a^{4}-b \,a^{3}+b^{2} a^{2}+a \,b^{3}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2}+b \,a^{3}-b^{2} a^{2}-a \,b^{3}+b^{4}\right ) \tan \left (\frac {\lambda x}{2}\right )^{2}+\left (-a^{4}-b \,a^{3}+b^{2} a^{2}+a \,b^{3}\right ) \sin \left (\frac {\lambda x}{2}\right )^{2}+\left (4 \sqrt {a^{2}-b^{2}}\, a^{2} b +4 \sqrt {a^{2}-b^{2}}\, a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+\left (-2 \sqrt {a^{2}-b^{2}}\, c_{1} a^{2}-2 \sqrt {a^{2}-b^{2}}\, c_{1} a b \right ) \sin \left (\lambda x \right )+\left (a^{4}-b \,a^{3}-b^{2} a^{2}+a \,b^{3}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2}+b \,a^{3}-b^{2} a^{2}-a \,b^{3}+b^{4}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (-a -b +\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )^{2}\right ) \left (a \sqrt {a^{2}-b^{2}}\, \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2} a b +\cos \left (\lambda x \right ) c_{1} a +\left (b a -b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )+c_{1} b \right )} \]
✓ Solution by Mathematica
Time used: 5.913 (sec). Leaf size: 184
DSolve[(a*Cos[\[Lambda]*x]+b)*(y'[x]-y[x]^2)-a*\[Lambda]^2*Cos[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\lambda \left (2 a b \sin (\lambda x) \coth ^{-1}\left (\frac {(a+b) \cot \left (\frac {\lambda x}{2}\right )}{\sqrt {(a-b) (a+b)}}\right )+\sqrt {a^2-b^2} (a c_1 \lambda (b-a) (a+b) \sin (\lambda x)+a \cos (\lambda x)-b)\right )}{2 b (a \cos (\lambda x)+b) \coth ^{-1}\left (\frac {(a+b) \cot \left (\frac {\lambda x}{2}\right )}{\sqrt {(a-b) (a+b)}}\right )-\sqrt {a^2-b^2} (a \sin (\lambda x)+c_1 \lambda (a-b) (a+b) (a \cos (\lambda x)+b))} \\ y(x)\to \frac {a \lambda \sin (\lambda x)}{a \cos (\lambda x)+b} \\ \end{align*}