61.11.10 problem 36

Internal problem ID [12210]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with tangent.
Problem number : 36
Date solved : Tuesday, January 28, 2025 at 01:10:57 AM
CAS classification : [_Riccati]

\begin{align*} x y^{\prime }&=a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m} \end{align*}

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 31

dsolve(x*diff(y(x),x)=a*tan(lambda*x)^m*y(x)^2+k*y(x)+a*b^2*x^(2*k)*tan(lambda*x)^m,y(x), singsol=all)
 

\[ y = -\tan \left (-a b \left (\int x^{k -1} \tan \left (\lambda x \right )^{m}d x \right )+c_{1} \right ) b \,x^{k} \]

✓ Solution by Mathematica

Time used: 0.557 (sec). Leaf size: 50

DSolve[x*D[y[x],x]==a*Tan[\[Lambda]*x]^m*y[x]^2+k*y[x]+a*b^2*x^(2*k)*Tan[\[Lambda]*x]^m,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^xa K[1]^{k-1} \tan ^m(\lambda K[1])dK[1]+c_1\right ) \]