61.8.10 problem 19

Internal problem ID [12170]
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 : 19
Date solved : Tuesday, January 28, 2025 at 12:47:02 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 24

dsolve(x*diff(y(x),x)=a*x^n*(y(x)+b*ln(x))^2-b,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.438 (sec). Leaf size: 35

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

\begin{align*} y(x)\to -b \log (x)+\frac {n}{-a x^n+c_1 n} \\ y(x)\to -b \log (x) \\ \end{align*}