61.8.9 problem 18

Internal problem ID [12169]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 12:46:22 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 1.345 (sec). Leaf size: 70

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

\[ y(x)\to \sqrt {b^2} x^k \tan \left (\frac {a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^m(\lambda x) (-k \log (\lambda x))^{-m} \Gamma (m+1,-k \log (x \lambda ))}{k}+c_1\right ) \]