Internal problem ID [9752]
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]
Solve \begin {gather*} \boxed {x y^{\prime }-a \,x^{2 n} \ln \relax (x ) y^{2}-\left (b \,x^{n} \ln \relax (x )-n \right ) y-c \ln \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 83
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)
\[ y \relax (x ) = \frac {\left (\tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (\ln \relax (x ) x^{n} b n +c_{1} n^{2}-b \,x^{n}\right )}{2 b^{2} n^{2}}\right ) \sqrt {4 b^{2} a c -b^{4}}-b^{2}\right ) x^{-n}}{2 a b} \]
✓ Solution by Mathematica
Time used: 1.234 (sec). Leaf size: 104
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+\sqrt {b^2-4 a c} \left (-1+\frac {2 c_1}{e^{\frac {x^n \sqrt {b^2-4 a c} (n \log (x)-1)}{n^2}}+c_1}\right )\right )}{2 a} \\ y(x)\to \frac {x^{-n} \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \\ \end{align*}