Internal problem ID [9768]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left (\sin \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )-\sin \left (\lambda x \right ) a \,\lambda ^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1223
dsolve((a*sin(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*sin(lambda*x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.338 (sec). Leaf size: 186
DSolve[(a*Sin[\[Lambda]*x]+b)*(y'[x]-y[x]^2)-a*\[Lambda]^2*Sin[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\lambda \left (2 a b \cos (\lambda x) \text {ArcTan}\left (\frac {a+b \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {b^2-a^2}}\right )+\sqrt {b^2-a^2} (a c_1 \lambda (b-a) (a+b) \cos (\lambda x)-a \sin (\lambda x)+b)\right )}{-2 b (a \sin (\lambda x)+b) \text {ArcTan}\left (\frac {a+b \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {b^2-a^2}}\right )+\sqrt {b^2-a^2} (-a \cos (\lambda x)+c_1 \lambda (a-b) (a+b) (a \sin (\lambda x)+b))} \\ y(x)\to -\frac {a \lambda \cos (\lambda x)}{a \sin (\lambda x)+b} \\ \end{align*}