Internal problem ID [10498]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (n +1\right ) x^{n} y^{2}-a \,x^{n +1} \ln \left (x \right )^{m} y=-a \ln \left (x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 201
dsolve(diff(y(x),x)=-(n+1)*x^n*y(x)^2+a*x^(n+1)*(ln(x))^m*y(x)-a*(ln(x))^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-{\mathrm e}^{\int \frac {\ln \left (x \right )^{m} x^{n} a \,x^{2}-2 n -2}{x}d x} x^{n} x +\int \left (-x^{n} n \,{\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right ) x^{-n}}{x \left (\int \left (-x^{n} n \,{\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 5.364 (sec). Leaf size: 311
DSolve[y'[x]==-(n+1)*x^n*y[x]^2+a*x^(n+1)*(Log[x])^m*y[x]-a*(Log[x])^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-2 (n+1)} \left (c_1 (n+1) x^{n+1} \int _1^x\exp \left (\frac {a \Gamma (m+1,-((n+2) \log (K[1]))) \log ^m(K[1]) (-((n+2) \log (K[1])))^{-m}}{n+2}-(n+2) \log (K[1])\right )dK[1]+c_1 \exp \left (\frac {a \log ^m(x) (-((n+2) \log (x)))^{-m} \Gamma (m+1,-((n+2) \log (x)))}{n+2}\right )+(n+1) x^{n+1}\right )}{(n+1) \left (1+c_1 \int _1^x\exp \left (\frac {a \Gamma (m+1,-((n+2) \log (K[1]))) \log ^m(K[1]) (-((n+2) \log (K[1])))^{-m}}{n+2}-(n+2) \log (K[1])\right )dK[1]\right )} y(x)\to \frac {x^{-2 (n+1)} \left (\frac {\exp \left (\frac {a \log ^m(x) (-((n+2) \log (x)))^{-m} \Gamma (m+1,-((n+2) \log (x)))}{n+2}\right )}{\int _1^x\exp \left (\frac {a \Gamma (m+1,-((n+2) \log (K[1]))) \log ^m(K[1]) (-((n+2) \log (K[1])))^{-m}}{n+2}-(n+2) \log (K[1])\right )dK[1]}+(n+1) x^{n+1}\right )}{n+1} \end{align*}