Internal problem ID [9806]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 51.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \sin \left (\lambda x \right ) y^{2}-a \,x^{n} \cos \left (\lambda x \right ) y+x^{n} a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 124
dsolve(diff(y(x),x)=lambda*sin(lambda*x)*y(x)^2+a*x^n*cos(lambda*x)*y(x)-a*x^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\int \frac {x^{n} \left (\cos ^{2}\left (\lambda x \right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, a -2 \left (\cos ^{2}\left (\lambda x \right )\right ) \lambda +2 \lambda }{\sin \left (\lambda x \right ) \cos \left (\lambda x \right )}d x}}{\left (\int -\lambda \,{\mathrm e}^{\int \frac {x^{n} \left (\cos ^{2}\left (\lambda x \right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, a -2 \left (\cos ^{2}\left (\lambda x \right )\right ) \lambda +2 \lambda }{\sin \left (\lambda x \right ) \cos \left (\lambda x \right )}d x} \sin \left (\lambda x \right )d x \right ) c_{1}+1}+\frac {1}{\cos \left (\lambda x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+a*x^n*Cos[\[Lambda]*x]*y[x]-a*x^n,y[x],x,IncludeSingularSolutions -> True]
Not solved