Internal problem ID [9747]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \ln \relax (x )^{k} \left (y-b \,x^{n}-c \right )^{2}-b n \,x^{n -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
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)
\[ y \relax (x ) = -\frac {\left (-2 \ln \relax (x )^{k} x^{n} a b -2 \ln \relax (x )^{k} a c \right ) \ln \relax (x )^{-k}}{2 a}+\frac {1}{c_{1}-\left (\int \ln \relax (x )^{k} a d x \right )} \]
✓ Solution by Mathematica
Time used: 0.965 (sec). Leaf size: 44
DSolve[y'[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}{a \log ^{k+1}(x) E_{-k}(-\log (x))+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}