61.13.13 problem 59

Internal problem ID [12233]
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 : 59
Date solved : Tuesday, January 28, 2025 at 01:31:30 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 63

dsolve(diff(y(x),x)=lambda*sin(lambda*x)*y(x)^2+a*sin(lambda*x)*y(x)-a*tan(lambda*x),y(x), singsol=all)
 

\[ y = \frac {\operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) c_{1} a -1}{\cos \left (\lambda x \right ) \operatorname {Ei}_{1}\left (\frac {a \cos \left (\lambda x \right )}{\lambda }\right ) c_{1} a -{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{\lambda }} c_{1} \lambda -\cos \left (\lambda x \right )} \]

✗ Solution by Mathematica

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

DSolve[D[y[x],x]==\[Lambda]*Sin[\[Lambda]*x]*y[x]^2+a*Sin[\[Lambda]*x]*y[x]-a*Tan[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
 

Not solved