61.8.11 problem 20

Internal problem ID [12171]
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
Date solved : Tuesday, January 28, 2025 at 12:47:05 AM
CAS classification : [_Riccati]

\begin{align*} x y^{\prime }&=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 ) \end{align*}

✓ Solution by Maple

Time used: 0.061 (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)
 

\[ y = \frac {\left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b \left (n \ln \left (x \right )-1\right ) x^{n}+c_{1} n^{2}\right )}{2 b^{2} n^{2}}\right )-b^{2}\right ) x^{-n}}{2 a b} \]

✓ Solution by Mathematica

Time used: 0.907 (sec). Leaf size: 130

DSolve[x*D[y[x],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*}