Internal problem ID [10495]
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: 20.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x -a \,x^{2 n} \ln \left (x \right ) y^{2}-\left (b \,x^{n} \ln \left (x \right )-n \right ) y=c \ln \left (x \right )} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 80
dsolve(x*diff(y(x),x)=a*x^(2*n)*ln(x)*y(x)^2+(b*x^n*ln(x)-n)*y(x)+c*ln(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\left (b \left (n \ln \left (x \right )-1\right ) x^{n}+c_{1} n^{2}\right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} n^{2}}\right ) \sqrt {4 a \,b^{2} c -b^{4}}-b^{2}\right ) x^{-n}}{2 a b} \]
✓ Solution by Mathematica
Time used: 1.872 (sec). Leaf size: 130
DSolve[x*y'[x]==a*x^(2*n)*Log[x]*y[x]^2+(b*x^n*Log[x]-n)*y[x]+c*Log[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (-b+\frac {\sqrt {b^2-4 a c} \left (-e^{\frac {x^n \sqrt {b^2-4 a c} (n \log (x)-1)}{n^2}}+c_1\right )}{e^{\frac {x^n \sqrt {b^2-4 a c} (n \log (x)-1)}{n^2}}+c_1}\right )}{2 a} \\ y(x)\to \frac {x^{-n} \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \\ \end{align*}