Internal problem ID [10559]
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]
\[ \boxed {y^{\prime }-\lambda \sin \left (\lambda x \right ) y^{2}-a \,x^{n} \cos \left (\lambda x \right ) y=-x^{n} a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 122
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)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\int \frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \left (\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) x^{n} a +2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \lambda \tan \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{2}}d x}}{\left (\int -{\mathrm e}^{\int \frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \left (\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) x^{n} a +2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 \lambda x \right )}{2}}\, \lambda \tan \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{2}}d x} \sin \left (\lambda x \right ) \lambda d x \right ) c_{1} +1}+\frac {1}{\cos \left (\lambda x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+a*x^n*Cos[\[Lambda]*x]*y[x]-a*x^n,y[x],x,IncludeSingularSolutions -> True]
Not solved