Internal problem ID [9660]
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]
Solve \begin {gather*} \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}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 260
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)
\[ y \relax (x ) = \frac {\left (-x^{-n} x^{n} b^{2} n^{2}-2 x^{-n} x^{n} b \beta n -x^{-n} x^{n} \beta ^{2}+\tanh \left (\frac {\sqrt {b^{4} n^{4}+4 b^{3} \beta \,n^{3}-4 \alpha \,b^{2} \gamma \,n^{2}+6 b^{2} \beta ^{2} n^{2}-8 \alpha b \beta \gamma n +4 b \,\beta ^{3} n -4 \alpha \,\beta ^{2} \gamma +\beta ^{4}}\, \left (-\ln \relax (x ) b k n +\ln \left (a \,x^{k}+b \right ) b n -\ln \relax (x ) \beta k -c_{1} b k +\ln \left (a \,x^{k}+b \right ) \beta \right )}{2 k b \left (b^{2} n^{2}+2 b \beta n +\beta ^{2}\right )}\right ) \sqrt {b^{4} n^{4}+4 b^{3} \beta \,n^{3}-4 \alpha \,b^{2} \gamma \,n^{2}+6 b^{2} \beta ^{2} n^{2}-8 \alpha b \beta \gamma n +4 b \,\beta ^{3} n -4 \alpha \,\beta ^{2} \gamma +\beta ^{4}}\right ) x^{-n}}{2 \alpha \left (b n +\beta \right )} \]
✓ Solution by Mathematica
Time used: 4.711 (sec). Leaf size: 604
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} \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 ((b n+\beta ) \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}+3 b n+3 \beta \right )}{2 b k}\right )\right )-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4} \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}+3 b n+3 \beta \right )}{2 b k}\right )-c_1 (b n+\beta ) \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}+3 b n+3 \beta \right )}{2 b k}\right )+\sqrt {\alpha } \sqrt {\gamma } c_1 \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4} \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}+3 b n+3 \beta \right )}{2 b k}\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*}