Internal problem ID [10499]
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]
\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y+a b x \ln \left (x \right )^{n +1} y=b \ln \left (x \right )+b} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 53
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)
\[ y \left (x \right ) = \left (\int {\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} b \left (\ln \left (x \right )+1\right )d x +c_{1} \right ) {\mathrm e}^{\int \left (-\ln \left (x \right )^{1+n} a b x +a \ln \left (x \right )^{n}\right )d x} \]
✓ Solution by Mathematica
Time used: 0.86 (sec). Leaf size: 124
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]
\[ y(x)\to \exp \left (a 2^{-n-2} (-\log (x))^{-n} \log ^n(x) \left (b \Gamma (n+2,-2 \log (x))+2^{n+2} \Gamma (n+1,-\log (x))\right )\right ) \left (\int _1^xb \exp \left (-2^{-n-2} a \left (2^{n+2} \Gamma (n+1,-\log (K[1]))+b \Gamma (n+2,-2 \log (K[1]))\right ) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) (\log (K[1])+1)dK[1]+c_1\right ) \]