61.13.5 problem 51

Internal problem ID [12225]
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
Date solved : Tuesday, January 28, 2025 at 01:25:56 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-a \,x^{n} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 103

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)
 

\[ y = \frac {-c_{1} {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x}+\sec \left (\lambda x \right ) \lambda \left (\int {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x} \sin \left (\lambda x \right )d x \right ) c_{1} -\sec \left (\lambda x \right )}{\lambda \left (\int {\mathrm e}^{\int \left (\cos \left (\lambda x \right ) x^{n} a +2 \tan \left (\lambda x \right ) \lambda \right )d x} \sin \left (\lambda x \right )d x \right ) c_{1} -1} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],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