Internal problem ID [9746]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
Solve \begin {gather*} \boxed {y^{\prime }-a \ln \relax (x )^{n} y+a b x \ln \relax (x )^{n +1} y-\ln \relax (x ) b -b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 57
dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)-a*b*x*(ln(x))^(n+1)*y(x)+b*ln(x)+b,y(x), singsol=all)
\[ y \relax (x ) = \left (\int {\mathrm e}^{-a \left (\int \left (-\ln \relax (x )^{n +1} b x +\ln \relax (x )^{n}\right )d x \right )} b \left (\ln \relax (x )+1\right )d x +c_{1}\right ) {\mathrm e}^{\int \left (-\ln \relax (x )^{n +1} a b x +a \ln \relax (x )^{n}\right )d x} \]
✓ Solution by Mathematica
Time used: 0.508 (sec). Leaf size: 96
DSolve[y'[x]==a*(Log[x])^n*y[x]-a*b*x*(Log[x])^(n+1)*y[x]+b*Log[x]+b,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (a \log ^{n+1}(x) (b \log (x) E_{-n-1}(-2 \log (x))-E_{-n}(-\log (x)))\right ) \left (\int _1^xb \exp \left (a \log ^{n+1}(K[1]) (E_{-n}(-\log (K[1]))-b E_{-n-1}(-2 \log (K[1])) \log (K[1]))\right ) (\log (K[1])+1)dK[1]+c_1\right ) \\ \end{align*}