61.8.6 problem 15

Internal problem ID [12166]
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 : 15
Date solved : Tuesday, January 28, 2025 at 12:46:09 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end{align*}

✓ Solution by Maple

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

dsolve(diff(y(x),x)=a*(ln(x))^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.848 (sec). Leaf size: 44

DSolve[D[y[x],x]==a*(Log[x])^k*(y[x]-b*x^n-c)^2+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{-\int _1^xa \log ^k(K[2])dK[2]+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}