Internal problem ID [9743]
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]
\[ \boxed {y^{\prime }-a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}-b n \,x^{n -1}=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = -\frac {\left (-2 a \ln \left (x \right )^{k} x^{n} b -2 a \ln \left (x \right )^{k} c \right ) \ln \left (x \right )^{-k}}{2 a}+\frac {1}{c_{1} -\left (\int a \ln \left (x \right )^{k}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.999 (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) \operatorname {ExpIntegralE}(-k,-\log (x))+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}