Internal problem ID [10403]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 73.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x \left (a \,x^{k}+b \right ) y^{\prime }-\alpha \,x^{n} y^{2}-\left (\beta -a n \,x^{k}\right ) y=\gamma \,x^{-n}} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 138
dsolve(x*(a*x^k+b)*diff(y(x),x)=alpha*x^n*y(x)^2+(beta-a*n*x^k)*y(x)+gamma*x^(-n),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{-n} \left (\tanh \left (\frac {\left (\left (-b n -\beta \right ) \ln \left (a \,x^{k}+b \right )+\left (\left (b n +\beta \right ) \ln \left (x \right )+c_{1} b \right ) k \right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}}{2 k b \left (b n +\beta \right )^{2}}\right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}+\left (b n +\beta \right )^{2}\right )}{2 \alpha \left (b n +\beta \right )} \]
✓ Solution by Mathematica
Time used: 4.641 (sec). Leaf size: 663
DSolve[x*(a*x^k+b)*y'[x]==\[Alpha]*x^n*y[x]^2+(\[Beta]-a*n*x^k)*y[x]+\[Gamma]*x^(-n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (b \left (n \left (-\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 n \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )+\exp \left (-\frac {(b n+\beta ) \left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right )}{b k}\right ) \left (\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+\beta \right ) \left (-\exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 \left (\beta -\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}\right ) \exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )\right )}{2 \alpha \left (\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )+c_1 \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )\right )} \\ y(x)\to \frac {x^{-n} \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}-b n-\beta \right )}{2 \alpha } \\ \end{align*}