Internal problem ID [9875]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary
functions (but not containing their derivatives).
Problem number: 29.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\lambda \sin \left (\lambda x \right ) y^{2}-f \left (x \right ) \cos \left (\lambda x \right ) y+f \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 118
dsolve(diff(y(x),x)=lambda*sin(lambda*x)*y(x)^2+f(x)*cos(lambda*x)*y(x)-f(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\int \frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \left (f \left (x \right ) \sin \left (\lambda x \right ) \cos \left (\lambda x \right )+2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \lambda \tan \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{2}}d x}}{\left (\int -{\mathrm e}^{\int \frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \left (f \left (x \right ) \sin \left (\lambda x \right ) \cos \left (\lambda x \right )+2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \lambda \tan \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{2}}d x} \sin \left (\lambda x \right ) \lambda d x \right ) c_{1} +1}+\frac {1}{\cos \left (\lambda x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+f[x]*Cos[\[Lambda]*x]*y[x]-f[x],y[x],x,IncludeSingularSolutions -> True]
Not solved