Internal problem ID [10524]
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]
\[ \boxed {\left (\cos \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )=a \,\lambda ^{2} \cos \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 204
dsolve((a*cos(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*cos(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (2 \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )-2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )+\left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}\right ) \left (a +b \right ) \left (a -b \right )\right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 \left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) b \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, a \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )-2 c_{1} \left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 7.903 (sec). Leaf size: 202
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) \text {arctanh}\left (\frac {(b-a) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^2-b^2}}\right )+\sqrt {a^2-b^2} \left (-a c_1 \lambda \left (a^2-b^2\right ) \sin (\lambda x)+a \cos (\lambda x)-b\right )\right )}{2 b (a \cos (\lambda x)+b) \text {arctanh}\left (\frac {(b-a) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^2-b^2}}\right )+\sqrt {a^2-b^2} \left (b c_1 \lambda \left (a^2-b^2\right )+a c_1 \lambda \left (a^2-b^2\right ) \cos (\lambda x)+a \sin (\lambda x)\right )} \\ y(x)\to \frac {a \lambda \sin (\lambda x)}{a \cos (\lambda x)+b} \\ \end{align*}