Internal problem ID [9748]
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: 16.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \ln \relax (x )^{n} y^{2}-b \ln \relax (x )^{m} y-b c \ln \relax (x )^{m}+a \,c^{2} \ln \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 134
dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)^2+b*(ln(x))^m*y(x)+b*c*(ln(x))^m-a*c^2*(ln(x))^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int a \ln \relax (x )^{n} {\mathrm e}^{\int \left (-2 \ln \relax (x )^{n} a c +\ln \relax (x )^{m} b \right )d x}d x \right ) {\mathrm e}^{\int \left (2 \ln \relax (x )^{n} a c -\ln \relax (x )^{m} b \right )d x} c +c_{1} {\mathrm e}^{\int \left (2 \ln \relax (x )^{n} a c -\ln \relax (x )^{m} b \right )d x} c +1\right ) {\mathrm e}^{\int \left (-2 \ln \relax (x )^{n} a c +\ln \relax (x )^{m} b \right )d x}}{c_{1}+\int a \ln \relax (x )^{n} {\mathrm e}^{\int \left (-2 \ln \relax (x )^{n} a c +\ln \relax (x )^{m} b \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 2.116 (sec). Leaf size: 385
DSolve[y'[x]==a*(Log[x])^n*y[x]^2+b*(Log[x])^m*y[x]+b*c*(Log[x])^m-a*c^2*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \left (-b \log ^m(K[1])+a c \log ^n(K[1])-a y(x) \log ^n(K[1])\right )}{a b (m-n) (c+y(x))}dK[1]+\int _1^{y(x)}\left (\frac {\exp \left (b \Gamma (m+1,-\log (x)) (-\log (x))^{-m} \log ^m(x)-2 a c \Gamma (n+1,-\log (x)) (-\log (x))^{-n} \log ^n(x)\right )}{a b (m-n) (c+K[2])^2}-\int _1^x\left (-\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \log ^n(K[1])}{b (m-n) (c+K[2])}-\frac {\exp \left (b \Gamma (m+1,-\log (K[1])) (-\log (K[1]))^{-m} \log ^m(K[1])-2 a c \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \log ^n(K[1])\right ) \left (-b \log ^m(K[1])+a c \log ^n(K[1])-a K[2] \log ^n(K[1])\right )}{a b (m-n) (c+K[2])^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]